Learning In Economic Systems With Expectations Feedback

Lecture Notes in Economics and Mathematical Systems Founding Editors: M. Beckmann H.P. Künzi Managing Editors: Prof. Dr...

Author: Wenzelburger J.

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Lecture Notes in Economics and Mathematical Systems Founding Editors: M. Beckmann H.P. Künzi Managing Editors: Prof. Dr. G. Fandel Fachbereich Wirtschaftswissenschaften Fernuniversität Hagen Feithstr. 140/AVZ II, 58084 Hagen, Germany Prof. Dr.W. Trockel Institut für Mathematische Wirtschaftsforschung (IMW) Universität Bielefeld Universitätsstr. 25, 33615 Bielefeld, Germany Editorial Board: A. Basile, A. Drexl, H. Dawid, K. Inderfurth, W. Kürsten, U. Schittko

555

Jan Wenzelburger

Learning in Economic Systems with Expectations Feedback With 4 Figures

123

Jan Wenzelburger Bielefeld University Department of Business Administration and Economics Postfach 100 131 33501 Bielefeld Germany [email protected]

ISBN-10 3-540-24322-4 Springer Berlin Heidelberg New York ISBN-13 978-3-540-24322-9 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Satz: Camera ready by author Cover: Erich Kirchner, Heidelberg Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig SPIN 11375647

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F¨ ur Frauke

Preface

Recently more and more economists have abandoned the assumption that agents have perfect knowledge about the economic environment in which they live. The focus has been turned to scenarios in which agents’ views of the world may be erroneous. The reason for this development is that the task of specifying all necessary capabilities which would allow an agent to attain the required perfect knowledge from observational data has not yet been accomplished. This fact holds particularly true for non-linear models in which forecasts feed back into the evolution of endogenous variables. These notes introduce the concept of a perfect forecasting rule which provides best least-squares predictions during the evolution of an economic system and thus generates rational expectation equilibria. The framework for adaptive learning schemes which under suitable conditions converge to perfect forecasting rules is developed. It is argued that plausible learning schemes should aim at estimating a perfect forecasting rule taking into account the correct feedback structure of an economy. A link is provided between the traditional rational-expectations view and recent behavioristic approaches. The material of these notes covers a revised version of my Habilitationsschrift which was accepted by the Fakult¨ at f¨ ur Wirtschaftswissenschaften at Bielefeld University. My primary scientiﬁc indebtedness is to Volker B¨ ohm whose profound concern with dynamic models in economics inspired me to write these notes. I am grateful for the fruitful and creative atmosphere at the Lehrstuhl B¨ ohm which all members created throughout the past years. Marten Hillebrand, Jochen Jungeilges, Thorsten Pampel, Olaf Schmitz, and George Vachadze helped to improve the manuscript at various stages and provided numerous helpful and critical comments. I am indebted to Hans Gersbach and Ulrich Horst for pointing out to me important new aspects of the theory.

Bielefeld, July 2006

Jan Wenzelburger

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Economic Systems With Expectations Feedback . . . . . . . . . . . . 2.1 An Introductory Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The General Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Forecasting Rules With Finite Memory . . . . . . . . . . . . . . . . . . . . . 2.4 Consistent Forecasting Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Unbiased Forecasting Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 8 10 12 14 17

3

Adaptive Learning in Linear Models . . . . . . . . . . . . . . . . . . . . . . . 3.1 Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Unbiased Forecasting Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 No-Updating Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Linear MSV Predictors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Approximating Unbiased Forecasting Rules . . . . . . . . . . . . . . . . . 3.4 Convergence of an AML-Based Learning Scheme . . . . . . . . . . . . 3.5 Mathematical Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 On the Stability of Linear Systems . . . . . . . . . . . . . . . . . . 3.5.2 Proofs of Theorem 3.2 and Theorem 3.3 . . . . . . . . . . . . . .

23 24 28 30 34 37 39 45 45 47

4

Economic Models Subject to Stationary Noise . . . . . . . . . . . . . 4.1 Forecasting Rules for Stationary Noise . . . . . . . . . . . . . . . . . . . . . 4.1.1 Consistent Forecasting Rules . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Unbiased Forecasting Rules . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Existence of Unbiased Forecasting Rules . . . . . . . . . . . . . . . . . . . . 4.3 MSV Predictors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Random Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Mathematical Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 59 61 64 68 73 75 79

X

Contents

5

Nonparametric Adaptive Learning . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.1 The Method of Stochastic Approximation . . . . . . . . . . . . . . . . . . 86 5.2 A Basic Convergence Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.3 Some Probabilistic Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.4 Suﬃcient Conditions for Contractions . . . . . . . . . . . . . . . . . . . . . . 102 5.5 Mathematical Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.5.1 Sobolev Imbedding Theorem . . . . . . . . . . . . . . . . . . . . . . . . 108 5.5.2 Proofs of Section 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.5.3 Proofs of Section 5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6

Stochastic Exchange Economies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.1 A Model of Pure Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.2 The Existence of Perfect Forecasting Rules . . . . . . . . . . . . . . . . . 129 6.3 Perfect Foresight Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.4 Adaptive Learning of Perfect Forecasting Rules . . . . . . . . . . . . . . 134 6.5 Mathematical Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7

Heterogeneous Beliefs in a Financial Market . . . . . . . . . . . . . . . 139 7.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.2 Heterogeneous Beliefs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 7.3 Risk Premia and Reference Portfolios . . . . . . . . . . . . . . . . . . . . . . 147 7.4 Mean-Variance Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.5 Perfect Forecasting Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 7.5.1 Perfect Forecasting Rules for First Moments . . . . . . . . . . 152 7.5.2 Perfect Forecasting Rules for Second Moments . . . . . . . . 153 7.6 Dynamics Under Rational Expectations . . . . . . . . . . . . . . . . . . . . 156 7.7 Adaptive Learning With Heterogeneous Beliefs . . . . . . . . . . . . . . 161 7.7.1 The General Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 7.7.2 The Case With Two Mediators . . . . . . . . . . . . . . . . . . . . . . 162 7.8 Mathematical Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

1 Introduction

The interplay between realizations of economic variables and agents’ beliefs concerning their future development is a fundamental feature of evolving markets. In many economic situations, agents’ subjective assessments of the future have a signiﬁcant impact on the evolution of an economy. Adaptive learning schemes are a widely accepted concept to describe the way in which agents form and update beliefs on the basis of past observations. The original intention of introducing such learning schemes was to justify the rational expectations hypothesis as a long-run concept in which agents successfully eliminated all systematic forecast errors, cf. Blume & Easley (1982). This justiﬁcation was ﬁrst supported by Bray (1982), Bray & Savin (1986), Fourgeaud, Gourieroux & Pradel (1986), Marcet & Sargent (1989a), and others, who provide conditions for linear models under which adaptive learning schemes ﬁnd rational expectations equilibria. These techniques are based on the ordinary leastsquares (OLS) method. The current state of these techniques including their convergence properties are presented in the book by Evans & Honkapohja (2001). While many economists use OLS-based learning schemes when modelling learning agents, there are four open issues which have not been resolved yet. First, most of the convergence results are only of a local nature, so that initial guesses of the system’s parameters may have to be quite close to their true values in order to assure convergence. The available criteria that assure global convergence are often diﬃcult or even impossible to verify. Second, in view of policy applications an open issue is how to select between multiple rationalexpectations solutions criteria during the estimation process using economic criteria. Third, a systematic treatment of models which involve forecasts for more than two periods ahead of the current state has not yet been accomplished. With regard to overlapping generation models with multiperiod life cycles, this is unsatisfactory. Finally, and most importantly, OLS-based techniques are restricted to linear and linearized models. Indeed, in economies whose evolution is driven by non-linear maps, the problem of learning schemes that generate forecasts

2

1 Introduction

converging to rational expectations equilibria remains to a large extent unresolved. For the standard OLG model of pure exchange, for example, Bullard (1994) and Marcet & Sargent (1989b) showed that ordinary least-squares learning may generate forecasts which do not converge to perfect foresight equilibria of the system. Convergence is obtained only if initial parameter guesses are suﬃciently close to their true values. Similar observations have been made for adaptive schemes such as Bayesian learning, where agents update beliefs according to Bayes’s rule (cf. Blume & Easley (1993, 1998) and references therein), and schemes based on parametric estimation techniques as in Kuan & White (1994). The fact that adaptive learning schemes may often fail to localize the desired equilibria is one reason why many economists have abandoned the classical rational expectations hypothesis and replaced it by a behavioral assumption on how agents update beliefs. The focus has been directed towards the long-run outcomes of learning behavior, sometimes referred to as learning equilibria. The economic justiﬁcation for this ‘agent-based’ behavioral approach is that agents with limited statistical tools may be unable to identify systematic forecast errors and hence see no reason to revise their behavior. For the example in Bullard (1994), Sch¨ onhofer (1999, 2001) demonstrates that the long-run outcome of the system may be described by consistent expectations equilibria in the sense of Hommes & Sorger (1998) in which the associated forecast errors have vanishing sample means and conﬁrm the hypothesized autocorrelation structure. The plausibility of Sch¨ onhofer’s results is questioned by Jungeilges (2003) who shows that the underlying linearity hypothesis for applying an OLS-based learning scheme in a non-linear environment may be rejected within ﬁnite time already by means of techniques from linear statistics. Hence, applying OLSbased learning schemes in non-linear models may be inappropriate. So far only a few contributions, as Chen & White (1998), Chatterji & Chattopadhyay (2000), or Wenzelburger (2002) succeeded in providing global results for nonlinear models supporting the original intention of the learning literature. In contrast to Bullard’s ﬁndings, Wenzelburger (2002) shows how a boundedly rational forecasting agency in a stationary exchange economy may always ﬁnd the perfect foresight equilibria of the economy. However, all these convergence results are conﬁned to a rather narrow class of non-linear cobweb-type models in which the endogenous variables do not feed back into the economic law and which contain at most two-period-ahead forecasts. These notes are motivated by the four open issues mentioned at the beginning of this introduction. We seek a method for non-linear models for which convergence results can be obtained and which allows to treat multiple rational-expectations solutions in a systematic manner. The focus is on multivariate models with expectational leads of arbitrarily ﬁnite length. These describe economic scenarios in which forecasts about state variables further than one period ahead of the current state feed back into the system. Expectational leads are intrinsic in situations in which the planning horizon of

1 Introduction

3

agents’ intertemporal decisions is longer than two periods. They occur naturally in overlapping generations with multi-period life cycles, as in Auerbach & Kotlikoﬀ (1987) or Bullard & Duﬀy (2001), K¨ ubler & Polemarchakis (2004) or Geanakopolos, Magill & Quinzii (2004) as well as in in ﬁnancial markets models with long-lived assets, e.g., see Hillebrand (2003) or Hillebrand & Wenzelburger (2006a). In macroeconomics, expectational leads may be related to announcement eﬀects, describing the impact of possible future actions by an authority such as a government or a bank on the current state of an economy. Following previous work by B¨ ohm & Wenzelburger (1999, 2002, 2004), we distinguish between an economic law describing the basic market mechanism of an economy and a forecasting rule which models the way in which a forecasting agency forms expectations. The combination of both ingredients together with a model for the exogenous perturbations may be seen as a deterministic dynamical system in a random environment which is deﬁned explicitly and globally on the whole state space. The concept of an unbiased forecasting rule is generalized to non-linear stochastic models with expectational leads. Unbiased forecasting rules by deﬁnition provide best least-squares predictions conditional on the available information and thus rational expectations in the classical sense. We interpret unbiased forecasting rules as the generators for rational expectations equilibria and use them to characterize these equilibria. This result complements the literature on rational expectations which often presumes extraordinary capabilities of agents without making explicit how agents can obtain the desired degree of rationality. In fact, the focus of the traditional literature is on the existence of rational-expectations solutions for particular models rather than on the forecasting rules pertaining to the forecasts, thus taking a solutions approach rather than a dynamical-systems approach, e.g., see Evans & Honkapohja (2001). The concept of a forecasting rule together with the notion of unbiasedness provides a link between the traditional rational-expectations view and recent behavioristic approaches. A key feature of systems with expectational leads is the fact that several forecasts, each formed at a diﬀerent time, refer to the same realization of an endogenous variable. This property turns out to be the primary cause for an inherent multiplicity of unbiased forecasting rules and thus for the multiplicity of rational expectations equilibria. Extending B¨ ohm & Wenzelburger (2002), we shed new light on this phenomenon by showing that an invertibility condition of the economic law is responsible for the existence and uniqueness of an unbiased forecasting rule. A special role is played by no-updating forecasting rules which are characterized by the property that forecasts formed at previous dates are never updated. We will show that unbiased no-updating forecasting rules yield the most precise forecasts in the sense that all forecasts are best least-squares predictions conditional on information available at a particular date. This includes those forecasts which were formed at a stage when this information was not yet available. This phenomenon is caused by the feedback property of the forecasts and constitutes a major diﬀerence between models with expectational leads and those without. In the linear case, the existence of

4

1 Introduction

an unbiased no-updating rule depends on the non-singularity of a particular parameter matrix, so that linear unbiased no-updating forecasting rules are analytically tractable whenever they exist. Having established the concept of an unbiased forecasting rule, the question is how agents can learn such a rule from historical data. The main methodological innovation of these notes is the systematic treatment of forecasts as exogenous inputs in the sense of systems and control theory (see, e.g., Caines 1988). At each point in time, we distinguish between the estimation of the system with an emphasis on the feedback of all relevant forecasts and the approximation of a desired forecasting rule. Four separate issues should thereby be distinguished carefully. First, the existence of a desired forecasting rule; second, the dynamic stability of the system under this particular forecasting rule; third, the dynamic stability of the system under the applied learning scheme and, fourth, the success of the learning scheme in terms of providing strongly consistent estimates of a forecasting rule. It will turn out that it is essentially the economic law which has to be estimated from time series data. The approach proposed in these notes opens up the possibility to select an approximation of a preferred forecasting rule on the basis of economic reasoning. It also allows, in principle, to keep a system dynamically stable at all stages of the learning scheme. The core of our adaptive learning scheme consists of nonparametric estimation techniques developed by Chen & White (1996, 1998, 2002) which are based on stochastic approximation methods of Yin & Zhu (1990) and Yin (1992). Nonparametric estimation techniques generalize the recursive ordinary least-squares algorithm which is well established in the learning literature. These techniques are indispensable when treating non-linear models without linearizing. Already Kuan & White (1994) and Chen & White (1998) incorporated the expectations feedback into the estimation of an economic system. However, it remains an open issue how to deal with the multiplicity of rational-expectations solutions in a more systematic manner. Since these multiplicities will occur generically, it is important to have a method that allows to select between multiple solutions on economic grounds rather than letting an algorithm decide which solution to select. In these notes the issue of specifying plausible learning schemes will, in essence, be reduced to a pure estimation and control problem. Roughly speaking, if the estimation technique succeeds in providing consistent estimators of the economic law, then the learning scheme will converge to the desired forecasting rule. Clearly, situations in which agents attain rational expectations are likely to remain special cases, as long as model speciﬁcations for which no consistent estimators exist are conceivable. A main conclusion, however, is that the proper incorporation of the expectations feedback is an indispensable methodological feature for the design of plausible learning schemes, as expectations, in general, will have a non-trivial feedback eﬀect on an economic system. It is left for future research to apply the methodology advocated in these notes to game theoretic settings and combine it with the theory of

1 Introduction

5

learning in games developed by Fudenberg & Levine (1998). An application to duopoly games, inspired by Kirman (1975) and Chen & White (1998), can be found in Wenzelburger (2004a) as a ﬁrst step in this direction. Most of the material of these notes does not have to be read in a chronological order. The general setup for non-linear economic models with expectational leads is introduced in Chapter 2. The basic methodological concepts are essentially contained in Chapter 3 which contains a nearly self-contained treatment of linear models and can be read without any knowledge about nonparametric methods. Chapter 4, which may be omitted in a ﬁrst reading, is primarily concerned with the issue of existence and uniqueness of unbiased forecasting rules and contains reﬁnements of Chapter 2. The nonparametric estimation techniques required for non-linear models are presented in Chapter 5. Those readers who are primarily interested in applications may start with either Chapter 6 or Chapter 7. As an example for a non-linear model, a standard version of a stochastic exchange economy is discussed in Chapter 6, a more involved ﬁnancial market model is presented in Chapter 7.

2 Economic Systems With Expectations Feedback

A fundamental feature of many economic systems is the interaction of a market mechanism with expectations of agents who try to predict the future evolution of the economy. As a way of formalizing this feedback eﬀect of expectations, we distinguish between an economic law describing the basic market mechanism of an economy and a forecasting rule which models the way in which an agent or a forecasting agency forms expectations. The combination of both ingredients with a model for exogenous perturbations constitutes a deterministic dynamical system in a random environment which describes the evolution of endogenous variables of an economy. Many well-known economic models can be cast in such a form. Powerful mathematical methods are available with the help of which their dynamic and stochastic behavior can be analyzed. The mathematical framework with many intuitive examples including iterated function systems (see Barnsley 1988) is, for example, provided in Lasota & Mackey (1994). More advanced methods are developed by Arnold (1998), Borovkov (1998) and others. In this chapter the non-linear setup introduced in B¨ ohm & Wenzelburger (1999, 2002, 2004) is extended to stochastic systems which involve forecasts for an arbitrarily ﬁnite period ahead of the current state of the system. The aim of these notes is to introduce a learning scheme which will ﬁnd the best least-squares predictions in such models. To this end we will ﬁrst introduce and classify those forecasting rules which provide the best least-squares predictions for a given economic scenario. We will show that these so-called unbiased forecasting rules together with the economic law are the generators of rational expectations equilibria. In Chapter 5 we will develop nonparametric methods to estimate unbiased forecasting rules from historical data. The distinction between an economic law, a forecasting rule, and a model for exogenous perturbations leads us to treating expectations systematically as exogenous inputs of an economic law. This concept will turn out to be important when establishing convergence results for learning schemes. Before introducing the general setup of these notes, let us start with a basic introductory example which stylizes a standard model in macroeconomics.

8

2 Economic Systems With Expectations Feedback

2.1 An Introductory Example Consider a univariate linear model of the form e e + B (0) yt,t+2 + ξt+1 , yt+1 = Ayt + B (1) yt,t+1

(2.1)

where yt ∈ R denotes the endogenous variable describing the state of the economy at date t, A, B (0) , and B (1) are non-random constants, and {ξt }t∈N e e is a sequence of exogenously given iid shocks. The variables yt,t+1 , yt,t+2 ∈R are forecasts for future realizations yt+1 and yt+2 , respectively, which are based upon information available at date t. Up to an additive constant which is omitted for simplicity, Evans & Honkapohja (2001, Chaps. 8 and 9) provide many well-founded economic examples which are of the functional form (2.1). In general, the model (2.1) admits multiple rational-expectations solutions, whereas the so-called minimal-state-variable solutions (MSV solutions) play a prominent role. An MSV solution, introduced by McCallum (1983), is an AR(1) process of the form (2.2) yt+1 = β yt + ξt+1 , where, according to the method of undetermined coeﬃcients, β is a solution of the quadratic equation 0 = A + (B (1) − 1)β + B (0) β 2 .

(2.3)

Clearly, MSV solutions are economically meaningful only for real β , i.e., for (1 − B (1) )2 ≥ 4AB (0) and, as is well known, they are generically non-unique. We will show in Example 2.2 below, any real solution β of (2.3) deﬁnes a forecasting rule e e yt,t+1 = β yt , yt,t+2 = β2 yt (2.4) which is unbiased in the sense that it provides best least-squares predictions for the system (2.1) when the forecasts (2.4) are inserted. That is to say, the process (2.2) generates rational expectations equilibria of the system (2.1). Imagine now a forecasting agency in charge of issuing forecasts without knowing the true model (2.1). Their ﬁrst task will be to learn the rationalexpectations solutions of the model by means of an adaptive learning scheme. The second task is to select between multiple such solutions. For example, if (1) (2) (1) (2) (2.1) admits two real MSV solutions β and β with |β | < 1 < |β |, (1) it may be desirable to select the non-explosive one β . Depending on the economic context, other solutions such as ARMA solutions might also be of particular interest, cf. Evans & Honkapohja (1986). One way of selecting a rational expectations solution is given by the expectational-stability criterion ﬁrst introduced by Evans (1985, 1986) and further developed in Evans (1989) and Evans & Honkapohja (1992). This criterion provides conditions under which a learning scheme converges to a particular rational-expectations solution. As has been noted by McCallum (1999), the problem with this approach

2.1 An Introductory Example

9

is that it is beyond the control of a forecasting agency to select a particular solution. In terms of the linear model (2.1), the methodological diﬀerence of the approach advocated in these notes in comparison to the traditional literature can be outlined as follows. The classical approach starts with a so-called perceived law of motion which, in view of possible MSV solutions, is of the form yt+1 = βyt + ξt+1 . (2.5) Then the method of ordinary least squares (OLS) is used to estimate the unknown coeﬃcient β in (2.5) from historical observations. This learning scheme is successful if the sequence of subsequent OLS-estimates {βˆt }t∈N for β converges to a MSV solution. Notice that there is no way to select between the two (1) (2) possible solutions β and β on the basis of economic reasoning. Instead, the convergence properties of the OLS scheme determine the outcome. Contrary to this traditional approach, the basic idea of our approach is to successively estimate the parameters (A, B (0) , B (1) ) of (2.1) from a perceived economic law ˆ t+B ˆ (1) y e ˆ (0) y e yt+1 = Ay t,t+1 + B t,t+2 + ξt+1

(2.6)

e which is of the very same form as (2.1), thereby treating the forecasts yt,t+1 e and yt,t+2 as exogenous inputs. These may be thought of as being obtained ˆt(0) , B ˆt(1) ) denote the from some previously applied forecasting rule. Let (Aˆt , B (0) period-t OLS-estimates of the unknown parameters (A, B , B (1) ) obtained from estimating the perceived law (2.6). Then the equation

ˆ (1) − 1)β + B ˆ (0) β 2 0 = Aˆt + (B t t

(2.7)

approximates equation (2.3) which deﬁnes MSV solutions. Whenever (2.7) ad(1) (2) mits positive real solutions βˆt and βˆt , these approximate the two possible MSV solutions. A major advantage of such an approach is that it allows to select an approximation of the preferred solution for each parameter estimate using economic criteria. As soon as the estimates are suﬃciently precise, a forecasting agency, for instance, would be in a position to choose a non-explosive solution to (2.1) whenever such a solution exists. Approximations of other ˆ t(0) , B ˆ t(1) ) as well. Moreover, a foresolutions could be computed from (Aˆt , B casting agency may want to alter the forecasts in order to keep the system dynamically stable. This methodology of estimating linear models with expectational leads will be treated in Chapter 3. In subsequent chapters, we will show that this methodology carries over to general non-linear models with expectational leads of arbitrary ﬁnite length. Such a generalization is necessary, for example, to treat overlapping generations models with multi-period life cycles, cf. Auerbach & Kotlikoﬀ (1987). The remainder of this chapter will therefore be devoted to the general non-linear setup used throughout these notes.

10

2 Economic Systems With Expectations Feedback

2.2 The General Setup Let X ⊂ Rd denote the space of all possible realizations of the endogenous variables and xt ∈ X be the vector of endogenous variables describing the state of the economy at date t. The vector of exogenous variables is denoted by ξt ∈ Ξ and assumed to be driven by an exogenous stochastic process. To ﬁx notation, we make the following assumption regarding this noise process. Assumption 2.1. The exogenous noise is driven by a sequence of random variables {ξt }t∈N on a probability space (Ω, F , P) with values in a subset Ξ ⊂ Rdξ , such that for each t ∈ N, ξt : Ω → Ξ. It is often convenient to assume that each state xt ∈ X can be subdivided into xt = (xt , yt ) ∈ X × Y = X, where yt ∈ Y ⊂ Rdy describes the endogenous ¯ variables for which expectations are formed, and xt ∈ X ⊂ Rdx is the vector of the remaining variables with dx = d¯x + dy . Let Ym = Y × . . . × Y . m − times denote the m-fold product of Y. An economic law is a map G : Ξ × X × Ym −→ X

(2.8)

with the interpretation that e e xt+1 = G(ξt+1 , xt , yt,t+1 , . . . , yt,t+m )

describes the state of the economy at time t + 1, given the predicted values e e yt,t+1 , . . . , yt,t+m ∈Y

formed at time t for yt+1 , . . . , yt+m , respectively, the current state xt , and a realization ξt+1 ∈ Ξ of the stochastic noise variable. The state xt itself may be a vector of lagged endogenous variables including past realizations ξs , s ≤ t of the exogenous noise process and past forecasts. We say that G has an expectational lead if it contains forecasts for more than one period ahead of the current state, that is, if m > 1. Throughout the remainder of these notes we assume m > 1 without further mentioning. The case m = 1 is extensively treated in B¨ ohm & Wenzelburger (2002). The deterministic map G represents the true economic system and determines the state of an economy one time step ahead of the current state, given subjective expectations and given the realization of the exogenous noise variable. From a stochastic viewpoint, each map G(ξt+1 (·), x, y e ) : Ω → X,

(x, y e ) ∈ X × Ym ,

t∈N

(2.9)

is a random variable on the probability space (Ω, F , P). In particular, given some perturbation ω ∈ Ω and some (xt , yte ) ∈ X × Ym , where

2.2 The General Setup

11

e e yte := (yt,t+1 , . . . , yt,t+m ) ∈ Ym

denotes the vector of current forecasts, the realization xt+1 (ω) = G(ξt+1 (ω), xt , yte ) of the random variable (2.9) describes the state of the economy in period t+1. It will be convenient to split the economic law (2.8) into two maps G : Ξ × X × Ym → X yielding

and g : Ξ × X × Ym → Y

xt+1 = G(ξt+1 , xt , yt , yte ) yt+1 = g(ξt+1 , xt , yt , yte )

.

(2.10)

In many applications it is assumed that agents form expectations for all relevant variables of the economic process under consideration, in which case Y = X and G ≡ g : Ξ × Xm+1 −→ X. e e , . . . , yt,t+m Remark 2.1. In many economic models the point forecasts yt,t+1 ∈ Y will be based on a subjective joint probability distribution of all future realizations yt+1 , . . . , yt+m and thus may be interpreted as subjective mean values for these realizations. An economic law of the form (2.8) could then be deﬁned as a function depending on subjective probability distributions rather than on point forecasts. For example, the overlapping generations model with multi-period life cycles and stochastic endowments ﬁts into such a framework, e.g., see K¨ ubler & Polemarchakis (2004) or Geanakopolos, Magill & Quinzii (2004). For the case of OLG models with two-period lives (m = 1), this has been done in B¨ ohm & Wenzelburger (2002). In focusing on expectational leads, we assume that subjective probability distributions are parameterized in subjective means and choose the functional form (2.8) for simplicity of exposition. The examples of Chapters 6 and 7 indicate how to generalize the setup of this chapter to economic laws which depend on subjective distributions. The fullﬂedged analysis of stochastic OLG models with multiperiod lives must be left for future research, cf. Hillebrand (2006). A linear OLG-model with expectational leads is analyzed in Hillebrand & Wenzelburger (2006a).

A complete description of the economy requires to specify a procedure e e , . . . , yt,t+m are determined in each period t. This of how the forecasts yt,t+1 speciﬁcation will depend on the type of information that agents have available at time t. Unless stated otherwise, we assume throughout these notes that a forecasting agency which is boundedly rational in the sense of Sargent (1993) is responsible for issuing all relevant forecasts. The forecasts are determined according to a forecasting rule (also referred to as a predictor) which at date t is assumed to be a function depending on all available information. Two cases will be treated. First, the case in which the economic system is governed by a Markov process with stationary transition probabilities. This

12

2 Economic Systems With Expectations Feedback

case, for example, arises naturally, if the exogenous noise {ξt }t∈N is an iid process. Second, the case in which the exogenous noise is driven by a stationary ergodic process. The ﬁrst case will be treated next, the second one is postponed to Chapter 4.

2.3 Forecasting Rules With Finite Memory For the remainder of this chapter, assume that the economic system is a homogeneous Markov process. In order to justify this assumption we need to specify the rules according to which forecasts are formed. At date t, the time series information of such a forecasting rule is comprised in the actual history e }s≤t , where of the economy {xs , ys−1 e e e = (ys−1,s , . . . , ys−1,s−1+m ) ∈ Ym , ys−1

s ∈ N.

(2.11)

To ﬁx the notation, let r ∈ N be arbitrary but ﬁxed and set Σ := (Ym × X) × · · · × (Ym × X) (r − 1) − times and

Σ := Σ × (Ym × X)

for the set of ﬁnite histories, relevant for the economy. Apart from the current state of the economy xt , let us assume that the relevant information in period t is given by an information vector of past observed states and forecasts e e , xt−1 , . . . , yt−r , xt−r ) ∈ Σ. Zt = (yt−1

(2.12)

Formally, a forecasting rule for an economic law could be an arbitrary function depending on the whole history of the economy. In view of a Markovian structure and in order to classify those forecasting rules which are capable of providing best least-squares predictions, however, we focus on a special class of functions that depend on a ﬁnite amount of past realizations given by (2.12). The information vector (2.12) could, in principle, include other quantities such as sunspot variables. However, in order to obtain best least-squares predictions along a possible time path, it will turn out that this information vector contains precisely the required time-series information at date t, provided that the length of the memory r is long enough. Letting (2.12) be the relevant information vector, a forecasting rule Ψ = (Ψ (1) , . . . , Ψ (m) ) is formally deﬁned as a function Ψ : X × Σ → Ym ,

(x, Z) → Ψ (x, Z),

such that e = Ψ (i) (xt , Zt ), yt,t+i

i = 1, . . . , m

(2.13)

2.3 Forecasting Rules With Finite Memory

13

e e describe the vector of forecasts yte = (yt,t+1 , . . . , yt,t+m ) in period t. If e m (xt , Zt , yt ) ∈ X × Σ × Y denotes the state of the system at date t, then the state of the system in period t + 1 is given by ⎧

xt+1 = G ξt+1 , xt , yte ) ⎪ ⎪ ⎨

Zt+1 = yte , xt , prΣ (2.14) −r Zt ⎪ ⎪ ⎩ e yt+1 = Ψ (xt+1 , Zt+1 )

with

prΣ −r : Σ → Σ ,

(Z (1) , . . . , Z (r) ) → (Z (1) , . . . , Z (r−1) )

denoting the projection onto the ﬁrst r − 1 components. Replacing yte with Ψ (xt , Zt ), the map (2.14) may be rewritten in the more compact form GΨ : Ξ × X × Σ → X × Σ, where

xt+1 = G ξt+1 , xt , Ψ (xt , Zt )

. Zt+1 = Ψ (xt , Zt ), xt , prΣ −r Zt )

(2.15)

Assuming that the forecasting agency uses the forecasting rule Ψ , the evolution of the economy is driven by the economic law G together with Ψ . Given a sequence of realizations of the exogenous noise process {ξt }t∈N and initial conditions (x0 , Z0 ) ∈ X × Σ, the sequence of states γ(x0 , Z0 ) := {(xt , Zt )}t∈N generated by (2.15) describes how the economy evolves over time. In the sequel, we will refer to γ(x0 , Z0 ) as an orbit of the system GΨ . To see when (2.15) is indeed a Markov process, deﬁne for each period t ∈ N the sub-σ-algebra

(2.16) Ft,r := σ (xs , yse ) t − r − 1 < s ≤ t of F which describes the relevant information of period t generated by the information vector (2.12). The whole history of the process at time t is represented by the sub-σ-algebra Ft := Ft,∞ . Let E[·|Ft,r ] denote the conditional expectations operator with respect to Ft,r and E[·|Ft ] denote the conditional expectations operator with respect to Ft . Then the assumption that (2.15) is a Markov process is formally stated as follows. Assumption 2.2. The length r of the information vector (2.12) is chosen suﬃciently large so that (2.15) is a homogeneous Markov process in the sense that E[xt+1 |Ft,r ] = E[xt+1 |Ft ] P − a.s. for all t ∈ N.

14

2 Economic Systems With Expectations Feedback

The factorization lemma (Bauer 1992, p. 71) and the choice of the forecasting rule Ψ in (2.14) imply the existence of a measurable map GE such that

(2.17) E[xt+1 |Ft,r ] = E G(ξt+1 , xt , yte )|Ft,r ] = GE (xt , Zt ) P − a.s. for each t ∈ N. Notice that each forecast yte is based on information available in period t and hence must be Ft measurable. Assumption 2.2 together with (2.17) thus justify the functional form of the forecasting rule (2.13). A special case arises when {ξt }t∈N is an iid process. Remark 2.2. If {ξt }t∈N is an iid process, then it is well known that

E[xt+1 |Ft ] = E G(ξt+1 , xt , yte ) = GE (xt , yte ) P − a.s. for all t ∈ N, cf. Bauer (1991, p. 134). In this case one may set Σ = Ym for the set of ﬁnite histories, r = 0 for the memory length, and Ft,0 = F for e all t ∈ N. Choosing the information vector of period t to be Zt = yt−1 , one obtains

e |Ft ] = E Ψ G(ξt+1 , xt , yte ), yte = ΨE (xt , yte ) P − a.s. E[yt+1 for some suitable function ΨE , so that (2.15) becomes the Markov process

xt+1 = G ξt+1 , xt , Ψ (xt , Zt ) . Zt+1 = Ψ (xt , Zt ) The justiﬁcation for this choice of Zt will become clear in Remark 2.4 below, after unbiased forecasting rules have been introduced. We are now ready to introduce the notion of an unbiased forecasting rule which is designed to provide best least-squares predictions for a given economic law for the case in which the system (2.14) is a homogeneous Markov process.

2.4 Consistent Forecasting Rules Before unbiased forecasting rules can be introduced, the notion of a consistent forecasting rule will be established in this section. A fundamental property of systems with expectational leads is that in each period t there are m forecasts for yt+1 of which m − 1 forecasts have been set prior to date t. This implies that the current forecast will in some sense have to be consistent with the previously formed forecasts. Otherwise not all forecasts can be correct. Consistent forecasting rules will now be designed to form consistent expectations in the sense that the law of iterated expectations is respected.

2.4 Consistent Forecasting Rules

15

As before, denote by E[·|Ft−i ] the conditional expectations operator with e respect to Ft−i . Each forecast yt−i,t+1 is based on information available in e period t − i and must be Ft−i measurable. The forecast errors yt+1 − yt−i,t+1 , i = 0, . . . , m − 1, vanish on average conditional on the respective σ-algebras Ft−i if and only if

e (2.18) ) Ft−i = 0 P − a.s. E (yt+1 − yt−i,t+1 for all i = 0, . . . , m−1. In other words, each of the forecasts satisfying (2.18) is a best least-squares prediction for yt+1 conditional on available information. Hence, a forecasting rule which satisﬁes condition (2.18) in all periods t is unbiased conditional on available information in the sense that all forecasts have the smallest possible forecast error. The law of iterated expectations states that

(2.19) E E[yt+1 |Ft ] Ft−i = E[yt+1 |Ft−i ] P − a.s. e for yt+1 for all times t ∈ N and all i ≥ 0. This implies that the forecast yt,t+1 chosen in period t has to satisfy the consistency conditions e e |Ft−i ] = yt−i,t+1 E[yt,t+1

P − a.s.

(2.20)

for all i = 1, . . . , m−1. Otherwise one of the forecasts in (2.20) cannot be a best least-squares prediction. Similarly, the law of iterated expectations implies for e , i = 1, . . . , m − 1, that the forecasts yt,t+i e e |Ft−1 ] = yt−1,t+i E[yt,t+i

P − a.s.

(2.21)

Clearly, if (2.21) holds for all forecasts in all periods prior to time t, then (2.20) holds as well. Hence (2.18) is automatically satisﬁed provided that the e is a best least-squares prediction, that is, if most recent forecast yt,t+1

e E (yt+1 − yt,t+1 ) Ft = 0 P − a.s. This shows that the notion of consistency is a basic prerequisite for an unbiased forecasting rule. The consistency conditions (2.21) can now be expressed in terms of forecasting rules Ψ = (Ψ (1) , . . . , Ψ (m) ) of the form (2.13) by making use of the Markov property of (2.15). Let Ej [·] ≡ E[·|Fj,r ] denote the conditional expectations operator with respect to the σ-algebra Fj,r , with r ∈ N as above. Since (2.15) generates a homogeneous Markov process, we have

e |Ft−1 ] = Et−1 Ψ (i) (xt , Zt ) P − a.s. E[yt,t+i for each i = 1, . . . , m − 1 and each t ∈ N. By the factorization lemma, there (i) exists a measurable map ΨE such that

16

2 Economic Systems With Expectations Feedback (i)

e E[yt,t+i |Ft−1 ] = ΨE (xt−1 , Zt−1 ) P − a.s.

(2.22)

for all i = 1, . . . , m − 1. Using (2.22), the consistency conditions (2.21) then take the form (i) e P − a.s. (2.23) ΨE (xt−1 , Zt−1 ) = yt−1,t+i for all i = 1, . . . , m − 1. The requirement (2.23) states that for each i = e 1, . . . , m − 1, the expected value of the new forecast yt,t+i conditional on Ft−1 e has to coincide with the respective old forecast yt−1,t+i = Ψ (i+1) (xt−1 , Zt−1 ). This motivates the following notion of a consistent forecasting rule. Deﬁnition 2.1. A forecasting rule Ψ = (Ψ (1) , . . . , Ψ (m) ) is called locally consistent if there exists a subset U ⊂ X × Σ, such that for each i = 1, . . . , m − 1, (i) Ψ (i+1) (x, Z) = ΨE (x, Z) for all (x, Z) ∈ U, (i)

with ΨE being deﬁned in (2.22). If U = X × Σ, then Ψ is called globally consistent. Deﬁnition 2.1 provides a minimum requirement for a forecasting rule of the form (2.13) in order to provide best least-squares predictions. A consistent forecasting rule is designed to satisfy (2.21) by construction on a possibly restricted subset U ⊂ X × Σ. It follows from the law of iterated expectations that otherwise (2.20) and hence (2.18) cannot be satisﬁed for all forecasts. Example 2.1. The simplest example of a consistent forecasting rule is a forecasting rule that never updates previous forecasts. Such a no-updating forecasting rule Ψ = (Ψ (1) , . . . , Ψ (m) ) is formally deﬁned by setting

Ψ (i) x, (z, Z ) := z (i+1) , (x, z, Z ) ∈ X × Ym × (X × Σ ) (2.24) for each i = 1, . . . , m − 1, where z = (z (1) , . . . , z (m) ) ∈ Ym . Ψ (m) may be an arbitrary function depending on the state x and the history Z = (z, Z ). e = If xt ∈ X is the current state of the economy at date t and yt−1 e e (yt−1,t , . . . , yt−1,t−1+m ) the vector of forecasts formed in the previous period e e t − 1, then the current vector of forecasts yte = (yt,t+1 , . . . , yt,t+m ) is

e e = yt−1,t+i , yt,t+i

i = 1, . . . , m − 1,

e yt,t+m = Ψ (m) (xt , Zt ).

Thus the forecasting rule (2.24) never updates forecasts formed at previous dates.

2.5 Unbiased Forecasting Rules

17

2.5 Unbiased Forecasting Rules We are now ready to develop the notion of an unbiased forecasting rule for the case in which the system (2.14) is a homogeneous Markov process. Unbiased forecasting rules are designed to provide best least-squares predictions, given the history of the process. They will also be referred to as perfect forecasting rules for ﬁrst moments. If applied, these forecasting rules provide correct ﬁrst moments along orbits of the system. In this sense, an unbiased forecasting rule Ψ together with the economic law G generates rational expectations equilibria (REE). Notice ﬁrst that the Markov property of the process (2.15) and the factorization lemma (Bauer 1992, p. 71) imply

(2.25) Et [yt+1 ] = Et g(ξt+1 , xt , yte ) = gE (xt , yte , Zt ) P − a.s. for some suitable measurable map gE . Given the state of the economy xt ∈ X and a vector of forecasts yte ∈ Ym , the forecast error for yt+1 is e e = g(ξt+1 , xt , yte ) − yt,t+1 , yt+1 − yt,t+1

while the forecast error conditional on information available at date t is

e e = gE (xt , yte , Zt ) − yt,t+1 P − a.s. (2.26) Et yt+1 − yt,t+1 Using (2.26), this gives rise to the deﬁnition of a (conditional) mean error function associated with the economic law G. Deﬁnition 2.2. Let G = (G, g) be an economic law. The (conditional) mean error function of G is a function EG : Ym × X × Σ → Rdy , deﬁned by

EG (z, x, Z) := gE x, z, Z − z (1) , (2.27) where z = (z (1) , . . . , z (m) ) and gE is deﬁned in (2.25). The error function takes a particularly simple form if the exogenous noise is iid. Remark 2.3. If {ξt }t∈N is an iid process, then

Et [yt+1 ] = E g(ξt+1 , xt , yte ) = gE (xt , yte )

P − a.s.

for all t ∈ N. In this case the error function takes the form EG (z, x) := gE (x, z) − z (1) , where z = (z (1) , . . . , z (m) ).

(x, z) ∈ X × Ym ,

18

2 Economic Systems With Expectations Feedback

Given an arbitrary state of the economy (xt , Zt ) ∈ X × Σ in some period t, the mean error function describes all possible mean forecast errors conditional on all information available at date t, regardless of which forecasting or learning rule has been applied. Given the state (xt , Zt ), the conditional forecast error (2.26) vanishes if and only if e Et [yt+1 − yt,t+1 ] = EG (yte , xt , Zt ) = 0

P − a.s.

(2.28)

The idea of an unbiased forecasting rule is now to ﬁnd a locally consistent forecasting rule Ψ : U ⊂ X × Σ → Ym with yte = Ψ (xt , Zt ) which satisﬁes

(2.29) EG Ψ (xt , Zt ), xt , Zt = 0 whenever (xt , Zt ) ∈ U. The existence of an unbiased forecasting rule is thus reduced to solving the implicit equation (2.28). More formally, the desired criterion for an unbiased forecasting rule is deﬁned as follows. Deﬁnition 2.3. Given an economic law G = (G, g), a consistent forecasting rule Ψ is called locally unbiased for G, if there exists an open subset U ⊂ X×Σ such that

EG Ψ (x, Z), x, Z = 0 for all (x, Z) ∈ U. A major problem with rational/unbiased expectations in non-linear systems will be that an orbit γ(x0 , Z0 ) of GΨ starting with unbiased forecasts in some (x0 , Z0 ) ∈ U may lose this property over time. Thus, if Ψ is a forecasting rule which is locally unbiased in a neighborhood U ⊂ X × Σ of (x0 , Z0 ), then an additional requirement that guarantees that the orbit stays in U is needed. Formally, a suﬃcient requirement for the existence of such unbiased orbits is that there exists a subset U ⊂ U which is forward invariant under GΨ , that is, for which GΨ (ξ, x, Z) ∈ U

for all (x, Z) ∈ U , ξ ∈ Ξ.

(2.30)

In this case, any orbit γ(x0 , Z0 ) that starts in (x0 , Z0 ) ∈ U will forever remain in U . Summarizing, an unbiased forecasting rule is deﬁned as follows. (1)

(m)

Deﬁnition 2.4. Given an economic law G, a function Ψ = (Ψ , . . . , Ψ ) is called an unbiased forecasting rule for G if there exists a subset U ⊂ X × Σ such that the following holds: (i) Consistency. For i = 1, . . . , m − 1, (i+1)

Ψ

(i)

(x, Z) = ΨE (x, Z)

(ii) Local unbiasedness. Ψ satisﬁes

EG Ψ (x, Z), x, Z = 0

for all (x, Z) ∈ U.

for all (x, Z) ∈ U.

2.5 Unbiased Forecasting Rules

19

(iii) Invariance. U is forward-invariant under GΨ , i.e., GΨ (ξ, x, Z) ∈ U

for all (x, Z) ∈ U, ξ ∈ Ξ.

The three conditions of Deﬁnition 2.4 imply that a forecasting rule Ψ is unbiased if and only if all orbits γ(x0 , Z0 ), (x0 , Z0 ) ∈ U of GΨ are unbiased e e orbits. That is, for each t ∈ N, the forecasts yt,t+1 , . . . , yt,t+m ∈ Y provide best least-squares predictions for yt+1 , . . . , yt+m , respectively. In this sense, unbiased forecasting rules generate rational expectations equilibria (REE). Example 2.2. Reconsider the linear economic law (2.1) of Section 2.1 as given by e e yt+1 = Ayt + B (1) yt,t+1 + B (0) yt,t+2 + ξt+1 , (2.31) where {ξt }t∈N is a sequence of iid perturbations with zero means. The error function associated with (2.31) is e e e e EG (yt,t+1 , yt,t+2 , yt ) = Ayt + (B (1) − 1)yt,t+1 + B (0) yt,t+2 .

(2.32)

Suppose B (0) = 0. Solving the condition !

e e EG (yt,t+1 , yt,t+2 , yt ) = 0 e for yt,t+2 , we obtain an unbiased no-updating forecasting rule which takes the form (compare with Example 2.1) (1) e e e = Ψ (yt , yt−1,t+1 ) := yt−1,t+1 , yt,t+1 (2.33)

(2) e e e yt,t+2 − Ayt . = Ψ (yt , yt−1,t+1 ) := B1(0) (1 − B (1) )yt−1,t+1

As an alternative, consider a forecasting rule of the form e yt,t+1 = Φ(1) (yt ) := βyt , e yt,t+2 = Φ(2) (yt ) := β 2 yt ,

(2.34)

which has already been introduced in Section 2.1. Now, the unknown coeﬃcient β may be obtained by inserting the forecasting rule (2.34) into the error function (2.32) and solving the equation !

EG (βy, β 2 y, y) = 0

for all y ∈ R

for the unknown parameter β. This equation is equivalent to the quadratic equation ! A + (B (1) − 1)β + B (0) β 2 = 0. (2.35) The solutions to (2.35) are given by

20

2 Economic Systems With Expectations Feedback (i)

β

1 − B (1) ± = 2B (0)

1 − B (1) 2B (0)

2 −

A , B (0)

i = 1, 2.

(2.36)

Clearly, only real solutions are economically meaningful which is the case for (1 − B (1) )2 ≥ 4AB (0) . As an alternative to the unbiased no-updating rule (2.33), we obtain an unbiased forecasting rule of the form (2.34). This rule is deﬁned by (i) e = β y t , yt,t+1 (2.37) (i)2 e = β y t , yt,t+2 (i)

with β , i = 1, 2 given in (2.36). It is straightforward to verify that (2.37) is consistent in the sense of Deﬁnition 2.1. Indeed the forecasting rule (2.37) satisﬁes (i)

(i)

(i)2

e e e Et−1 [yt,t+1 ] = β Et−1 [yt ] = β yt−1,t = β yt−1 = yt−1,t+1

for all times t. A forecasting rule of the form (2.34) is referred to as a minimalstate-variable predictor (MSV predictor) as it uses a minimum amount of variables. We see from this example that already linear models may admit multiple unbiased forecasting rules. The existence and uniqueness of unbiased forecasting rules for general linear models and methods to estimate these from historical data will be addressed in Chapter 3. See in particular Theorem 3.1. We conclude this section with discussing the problem of existence of unbie ased no-updating forecasting rules. Since all forecasts yt,t+i , i = 1, . . . , m − 1 have to satisfy the above consistency conditions, the only forecast which is not e constrained by the consistency conditions is the forecast yt,t+m for the most leading variable yt+m . In the case with no-updating the ﬁrst m − 1 forecasts will be set to the values of the previous period such that i = 1, . . . , m − 1.

e e yt,t+i = yt−1,t+i ,

(2.38)

Given the information vector Zt , the idea now is to ﬁnd a forecasting rule (m)

e yt,t+m = Ψ

(xt , Zt )

and a subset U ⊂ X × Σ so that

e (m) e , . . . , yt−1,t+m−1 , Ψ (xt , Zt ), xt , Zt = 0 EG yt−1,t+1

(2.39)

whenever (xt , Zt ) ∈ U. The economic implication for such an unbiased noe has to be chosen in updating forecasting rule is that the newest forecast yt,t+m e e such a way that the previous forecast yt−1,t+1 = yt,t+1 becomes correct. Under e , i = 1, . . . , m − 1 become best no updating, then all previous forecasts yt−i,t+1

2.5 Unbiased Forecasting Rules

21

least-squares predictions for yt+1 conditional on information available at date t. Indeed, since by construction (2.38) all forecasts for one variable coincide, (2.39) implies e ] = 0 P − a.s. (2.40) Et [yt+1 − yt−i,t+1 for all i = 0, . . . , m − 1. Thus all previous forecasts inherit the precision of the e most recent forecast yt,t+1 . Remark 2.4. If {ξt }t∈N is an iid process, then the unbiased forecasting rule takes a special form. From Remark 2.3 we know that the error function reduces to EG (z, x) = gE (x, z) − z (1) , (x, z) ∈ X × Ym , e where z = (z (1) , . . . , z (m) ). The information vector of period t is Zt = yt−1 (m) and the m-th component function Ψ of an unbiased no-updating forecasting (m) rule Ψ is formally deﬁned by Ψ : X × Ym → Y. Condition (2.39) for the unbiasedness reads

e (m) e e e , . . . , yt−1,t−1+m , Ψ (xt , yt−1,t+1 , . . . , yt−1,t−1+m ), xt = 0 EG yt−1,t+1 (m)

e whenever (xt , yt−1 ) ∈ U. This demonstrates that the functional form of Ψ (m) is prescribed by the Implicit Function Theorem. In particular, Ψ must be (m) e so that Ψ : X × Ym−1 → Y. This justiﬁes the choices independent of yt−1,t made in Remark 2.2.

We see from (2.39) that the existence problem of unbiased forecasting rules is, in essence, reduced to the Implicit Function Theorem which determines the functional form of an unbiased forecasting rule. In the Markovian case of this section, the existence theorem for the deterministic case with expectational leads can readily be applied, cf. B¨ ohm & Wenzelburger (2004). Existence and uniqueness of unbiased no-updating forecasting rules will be established in Theorem 4.2 of Chapter 4 in which we generalize and reﬁne our framework to economic models which are perturbed by stationary ergodic noise. Assuming that an unbiased no-updating forecasting rule exists and exploiting the presumed Markov structure, the last ﬁnding of this chapter is summarized in the following theorem. Theorem 2.1. Assume that under the hypothesis of this chapter, an unbiased no-updating forecasting rule for the economic law (2.8) in the sense of Deﬁe nition 2.4 exists. Then for each t ∈ N, all forecasts yt−i,t+1 , i = 0, . . . , m − 1 for yt+1 are best least-squares prediction conditional on the history Ft , that is,

e E (yt+1 − yt−i,t+1 ) Ft = 0 P − a.s. for all i = 0, . . . , m − 1.

22

2 Economic Systems With Expectations Feedback

A remarkable fact in economic systems with expectational leads is that unbiased no-updating forecasting rules, if they exist, generate rational expece e tations in the sense that all forecasts yt−m+1,t+1 , . . . , yt,t+1 are best leastsquares predictions for yt+1 conditional on information available at date t, including those which have been made prior to that date. Thus, an unbiased no-updating rule yields the most precise forecasts in the sense that forecast errors vanish conditional on information which is not available at the stage in which they were issued. This peculiar phenomenon is caused by the feedback property of forecasts and indeed holds true only for forecasts which feed back non-trivially into the economic law.

Concluding Remarks The careful distinction between an economic law and a forecasting rule was the necessary step to obtain a proper description of an economic dynamical system in a Markovian environment. The results of this chapter demonstrate that the property of rational expectations along orbits of an economic dynamical system imposes structural restrictions on the economic fundamentals which are embodied in an economic law. A remarkable phenomenon in economic systems with expectational leads is the notion of an unbiased no-updating forecasting rule. Within the class of possible unbiased forecasting rules, these yield the most precise forecasts as forecasts are best least-squares predictions conditional on information which is not available at the stage in which they were issued. An economic implication of this chapter is that agents who know an unbiased forecasting rule also know how their forecasts inﬂuence the economy. In principle, these agents could try to strategically exploit this information. How this could be achieved is an interesting avenue for future research.

3 Adaptive Learning in Linear Models

In many economic applications linear models are popular either as exact formulations or as linear approximations of originally nonlinear speciﬁcations. For this reason it is worthwhile to devote a chapter to the study of the linear case. The advantage of linear models is that they do not require nonparametric estimation techniques, so that we can focus entirely on the methodology. In providing a benchmark situation for the general nonlinear case, it will be seen in Chapter 5 that the methodology for linear models carries over to the nonlinear situation. The main diﬀerence between the adaptive learning scheme developed in this chapter and the traditional approach of the learning literature (e.g., see Kirman & Salmon (1995) or Evans & Honkapohja 2001) is that forecasts are systematically treated as exogenous inputs in the sense of the systems and control literature. At each point in time, we distinguish between the estimation of the economic law and the application of an approximated unbiased forecasting rule to the system. The main advantage of this approach is that the system can be kept dynamically stable at all stages of the learning procedure. It opens up the possibility to select a particular preferred forecasting rule on the basis of an adequate economic reasoning, as soon as parameter estimates are suﬃciently precise. This, in essence, resolves an issue put forward in McCallum (1999). Our adaptive learning scheme for linear stochastic models is based on the recursive extended least-squares algorithm which is well established in the engineering literature, cf. Caines (1988). This algorithm is an extension of the popular ordinary least-squares (OLS) algorithm to a class of more general noise processes. Using ideas from the literature on adaptive control and optimal tracking, results of several papers by Lai and Wei, notably their 1986b paper, will be applied here. These authors pursue a martingale approach paralleling and extending the original approach of Ljung (1977) who uses a limiting non-random ordinary diﬀerential equation (ODE) for his convergence analysis.

24

3 Adaptive Learning in Linear Models

This chapter reports on reﬁnements of Wenzelburger (2006). We relax standard assumptions of the traditional learning literature as given in Evans & Honkapohja (2001) in several respects. First, we admit expectational leads of arbitrary length. Second, we generalize the class of stochastic processes which governs the exogenous observable variables, where, in addition, we do not assume that the process is known. Third, we admit colored noise for the unobservable stochastic perturbations acting on the system. This generalizes the usual white-noise assumption and seems to be well known in the literature on adaptive control, cf. Caines (1988). Finally, we show convergence of our learning scheme globally for all initial conditions under simple and standard identiﬁability conditions. These conditions are generically satisﬁed in the white-noise case. Thus well-known convergence results of the learning literature are strengthened because the conditions require relatively mild technical prerequisites and are much easier to verify and interprete than those based on Ljung’s ODE approach.

3.1 Linear Models Consider a linear economic law of the form yt =

n1 i=1

A(i) yt−i +

n2 i=0

e B (n2 −i) yt−1,t+i +

n3 i=1

C (i) wt−i + t +

n4

D(i) t−i + b,

i=1

(3.1) where yt ∈ Rd denotes a vector of endogenous variables describing the state of the economy at date t. A(i) , B (i) , C (i) , and D(i) are non-random d×d matrices and b ∈ Rd is a constant non-random vector. The exogenous perturbations are prescribed by means of two stochastic processes {wt }t∈N and {t }t∈N on a probability space (Ω, F, P) which are adapted to a ﬁltration {Ft }t∈N . {wt }t∈N describes the process of Rd -valued exogenous variables, which is assumed to be observable. The process {t }t∈N is an Rd -valued martingale diﬀerence sequence, i.e. E[t+1 |t , . . . , 0 ] = 0, t ∈ N, describing the unobservable perturbae tions which aﬀect the system (3.1). yt−1,t+i ∈ Rd , i = 0, . . . , n2 , are forecasts for future realizations yt+i , i = 0, . . . , n2 , respectively. Remark 3.1. A complete treatment of models of the type (3.1) seems as yet unaccomplished, cf. Evans & Honkapohja (2001, p. 173). Most models of the literature have n2 ≤ 1. The ﬁnancial market model discussed in Chapter 7, Section 7.4 may serve as an example for n2 = 1. This model is generalized in Hillebrand & Wenzelburger (2006a) to a ﬁnancial market model with traders whose planning horizon is longer than two periods and which is described by an economic law of the form (3.1) with arbitrarily large n2 > 1. The model (3.1) provides a quite general speciﬁcation of a linear economic system, where for the sake of simplicity we abstract from lagged forecasts

3.1 Linear Models

25

made at previous dates as well as from contemporaneous forecasts. For the purposes of this chapter, it is convenient to reformulate the model (3.1) as follows. To this end we adjust the notation of Chapter 2 to the particular needs of this chapter and use ‘ ’ to distinguish between row and column vectors. This also facilitates further reading of the related literature. For each t ∈ N, set Yt := (yt , . . . , yt−n ) , 1 +1

e e Yte := (yt,t+1+n , . . . , yt,t+1 ) 2

Wt := (wt , . . . , wt−n ) , 3 +1

Et := ( t , . . . , t−n4 +1 )

and

xt = (Yt , Wt , Et ) .

(3.2)

Let Id denote the identity matrix on Rd and deﬁne block matrices associated with (3.1) as ⎞ ⎛ (1) ⎞ ⎛ (0) A · · · · · · A(n1 ) B · · · B (n2 ) ⎜ Id 0 · · · 0 ⎟ ⎜ 0 ··· 0 ⎟ ⎜ ⎟ ⎟ ⎜ A := ⎜ . . . , B := ⎟ ⎜ .. .. ⎟ , . . . . ... ⎠ ⎝ .. ⎝ . . ⎠ 0 · · · Id

⎞ C (1) · · · C (n3 ) ⎜ 0 ··· 0 ⎟ ⎟ ⎜ C := ⎜ . .. ⎟ , ⎝ .. . ⎠ 0 ··· 0 ⎛

0 ···

0

0

⎞ D(1) · · · D(n4 ) ⎜ 0 ··· 0 ⎟ ⎟ ⎜ D := ⎜ . .. ⎟ . ⎝ .. . ⎠ 0 ··· 0 ⎛

(3.3)

Let prn−1 : Rdn → Rdn , n ∈ N, be a projection mapping, given by ⎛ ⎞ 0 ··· ··· 0 ⎜ Id 0 · · · 0 ⎟ ⎜ ⎟ pr n−1 := ⎜ . . . . ⎟ , n ∈ N. ⎝ .. . . . . .. ⎠ 0 · · · Id 0 A linear economic law with expectational leads is an (aﬃne-)linear map G : R2d × Rd(n1 +n3 +n4 ) × Rd(n2 +1) → Rd(n1 +n3 +n4 ) , deﬁned by xt+1 = G(ξt+1 , xt , Yte ) (3.4) ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ e BYt (t+1 + b , 0, . . . , 0) A C D 3 0 ⎠ xt + ⎝ 0 ⎠ + ⎝ (wt+1 , 0, . . . , 0) ⎠ , := ⎝ 0 prn−1 n4 0 0 pr−1 0 (t+1 , 0, . . . , 0) 2d with ξt+1 = ( t+1 , wt+1 ) ∈ R . The linear speciﬁcation of the economic law (3.1) suggests to restrict the analysis of forecasting rules to (aﬃne-) linear functions Ψ : Rd(n1 +n3 +n4 ) × Rd(n2 +1) → Rd(n2 +1) , given by

26

3 Adaptive Learning in Linear Models e Yte := Ψ (xt , Yt−1 ),

(3.5)

such that Yte is the vector forecasts formed at date t ∈ N. It will turn out that forecasting rules which generate rational expectations equilibria (REE) of the model (3.1) are of the functional form (3.5). In order to actually apply (3.5), we will assume for the moment that, contrary to the assumptions stated at the outset, {s }ts=0 are observable in period t. When introducing our learning scheme in Section 3.3 below, we will replace the {t }t∈N with appropriate estimates. Given the economic law G and a forecasting rule Ψ , the future state of the economy is determined by the time-one map GΨ (ξt+1 , ·) : Rd(n1 +n3 +n4 ) × Rd(n2 +1) → Rd(n1 +n3 +n4 ) × Rd(n2 +1) ,

(3.6)

ξt+1 ∈ R2d , deﬁned by

e G ξt+1 , xt , Ψ (xt ,Yt−1 xt ) → . e e Yt−1 Ψ xt , Yt−1 e are the In other words, if xt is the (lagged) state of the economy at date t, Yt−1 previous forecasts and ξt+1 is the future realization of the exogenous shock, xt+1 e = GΨ (ξt+1 , xt , Yt−1 ) Yte

describes the state xt+1 of the economy at date t + 1 together with all relevant forecasts Yte . The map (3.6) thus drives the evolution of the economy. The system (3.6) can also be seen as a state-space representation of an ARMAX process, cf. Hannan & Deistler (1988). Other forecasting rules, in particular those which depend explicitly on time, can be implemented in a similar manner. This, in fact, will be done when introducing our learning scheme. Before we address the problem of learning REE from historical data, we state conditions under which the system (3.4) is stable in the sense that the growth of the realizations yt is P-a.s. bounded. This will become important when introducing our learning scheme because it guarantees dynamic stability of the economic system. Recall the following notation. By at = O(bt ) P-a.s. for two sequences of random numbers {at } and {bt }, we mean that sup abtt ≤ t∈N

c P-a.s. for some positive constant c. By at = o(bt ) P-a.s., we mean that lim abtt = 0 P-a.s.

t→∞

Proposition 3.1. Consider the economic law (3.1) and let Xt = (Yt , Yte , Wt , Et )

(3.7)

Assume that the following hypotheses are satisﬁed: (i) All eigenvalues λi of A given in (3.3) lie on or within the unit circle, that is, |λi | ≤ 1.

3.1 Linear Models

27

(ii) The two exogenous sequences {wt }t∈N and {}t∈N satisfy wt = o(tβ ) and = o(tβ ) P-a.s. for some real number β. If, in addition, the forecasts fulﬁll P-a.s. e

yt,t+i

= o(tβ )

for all j = 1, . . . , n2 + 1,

(3.8)

then the system (3.1) is stable in the sense that there exists α > 0 such that

Xt = O(tα )

P − a.s.

Corollary 3.1. Under the hypotheses of Proposition 3.1, if all eigenvalues λi of A given in (3.3) lie within the unit circle, i.e. |λi | < 1, then

Xt = o(tβ )

P − a.s.

The proofs of Proposition 3.1 and Corollary 3.1 are given in Section 3.5.1 below. Observe that Proposition 3.1 makes no assumption on how the forecasts have been obtained. In particular, the assumptions of Proposition 3.1 do not rule out that (3.1) admits explosive REE. The major implication of e Proposition 3.1 is that a forecasting agency controlling the forecasts yt,t+j , which are actually inserted into the economic law (3.4), has to ensure that these are bounded in the above sense in order to keep the economic system dynamically stable. If Condition (i) were violated, stability cannot be ensured without a-priori knowledge on the parameter matrices A and B. Remark 3.2. We brieﬂy comment on the case when Condition (i) of Proposition 3.1 on the eigenvalues of A is violated. Using the notation introduced above, an aﬃne-linear forecasting rule (3.5) is of the form e e ) := K Yt + L Yt−1 + M Wt + N Et + c, Yte = Ψ (Yt , Wt , Et , Yt−1

where K, L, M, and N are non-random matrices of respective dimensions and c ∈ Rd(n2 +1) is a non-random vector. Summarizing the exogenous processes acting on the system in ηt+1 := CWt + DEt + t+1 + b, ζt := MWt + NEt + c, the time-one map (3.6) takes the form Yt+1 := AYt + BYte + ηt+1 e Yte := KYt + LYt−1 + ζt or, in block-matrix representation, A + BK BL Yt Yt+1 Bζt + ηt+1 = + . e Yte Yt−1 ζt K L

(3.9)

28

3 Adaptive Learning in Linear Models

In view of Proposition 3.1, the major requirement for the stability of (3.9) is that all eigenvalues of the coeﬃcient matrix A + BK BL (3.10) K L lie within the unit circle. Given A and B, it is well known from control theory that K exists such that all eigenvalues of A + BK lie within the unit circle if the pair (A, B) is controllable, cf. Simon & Mitter (1968, Thm. 5.3, p. 325). Choosing L = 0, we see that in this case the realizations Yt can be made bounded, whereas the forecasts Yte are possibly unbounded. Except for the scalar case, choosing K and L such that (3.10) satisﬁes the eigenvalue condition is signiﬁcantly more diﬃcult. In any case, however, without a-priori knowledge on A and B, it is hardly possible to choose K and L so that the system (3.9) is stable. For this reason the conditions of Proposition 3.1 seem to be indispensable for adaptive learning schemes. Remark 3.3. In view of Chapter 4 we could assume that, in addition, the two processes {wt }t∈N and {t }t∈N admit a representation by a so-called ergodic metric dynamical system which we will introduce there. Then the time-one map (3.6) becomes an aﬃne random dynamical system in the sense of Arnold (1998).

3.2 Unbiased Forecasting Rules In this section we will illustrate the results of Chapter 2 by introducing an (aﬃne-)linear unbiased forecasting rule designed to generate rational expectations along all orbits of the linear system (3.1) or, in other words, rational expectations equilibria (REE). We will begin with linear no-updating rules and then discuss linear MSV predictors. To derive an unbiased linear forecasting rule, we brieﬂy review results e from Chapter 2. Recall that in each period t there are n2 forecasts yt−i,t+1 , i = 1, . . . , n2 , for yt+1 which have been set at dates prior to t. Denote by Ej [·] = E[·|Fj ] the conditional expectations operator. The forecast errors for yt+1 conditional on information available at the respective dates vanish if and only if e ] = 0 P − a.s., i = 0, . . . , n2 . (3.11) Et−i [yt+1 − yt−i,t+1 Condition (3.11) and the law of iterated expectations, i.e. Et−i [Et [yt+1 ]] = e e Et−i [yt+1 ], i ≥ 0, implies that all forecasts yt−n , . . . , yt,t+1 for yt+1 have 2 ,t+1 to satisfy the consistency conditions e e ] = yt−i,t+1 Et−i [yt,t+1

P − a.s.,

i = 1, . . . , n2 .

(3.12)

e e , . . . , yt,t+1 obey (3.12), then the On the other hand, if all forecasts yt−n 2 ,t+1 law of iterated expectations implies that (3.11) is automatically satisﬁed, proe has a vanishing conditional forecast vided that the most recent forecast yt,t+1

3.2 Unbiased Forecasting Rules

29

e error, that is, Et [yt+1 ]−yt,t+1 = 0 P-a.s. Since (3.12) has to hold for all times t e and analogously for all forecasts yt,t+j , we can transform (3.12) to consistency e e conditions for the ﬁrst n2 forecasts yt,t+1 , . . . , yt,t+n formed at date t. These 2 take the form e e ] − yt−1,t+i =0 Et−1 [yt,t+i

P − a.s.,

i = 1, . . . , n2 ,

(3.13)

e e , . . . , yt−1,t+n are the respective forecasts formed at date t − 1. where yt−1,t+1 2 In terms of linear forecasting rules Ψ = (Ψ (1) , . . . , Ψ (n2 +1) ) of the form (3.5), we have Ψ (i) : Rd(n1 +n3 +n4 ) × Rd(n2 +1) → Rd with e e := Ψ (i) (xt , Yt−1 ) yt,t+i

(3.14)

for each i = 1, . . . , n2 + 1. In view of Deﬁnition 2.1, Chapter 2, the consistency conditions (3.13) in terms of the linear forecasting rules (3.14) state that for each i = 1, . . . , n2 , e e )] = yt−1,t+i Et−1 [Ψ (i) (xt , Yt−1

P − a.s.

(3.15)

e ) and all periods t. As before, Condition (3.15) provides a minfor all (xt , Yt−1 imum requirement a forecasting rule has to meet in order to generate rational expectations, since otherwise (3.11) cannot be satisﬁed for all forecasts. To see which forecasting rules generate rational expectations, we ﬁrst compute the expected value of the future state yt conditional on information available at date t − 1 which is

Et−1 [yt ] =

n1 i=1

A(i) yt−i +

n2

e B (n2 −i) yt−1,t+i +

n3

C (i) wt−i +

D(i) t−i + b.

i=1

i=1

i=0

n4

e conditional on information available at The forecast error Et−1 [yt ] − yt−1,t date t − 1 is then obtained from the (mean) error function associated with the economic law (3.1) which is given by a function

EG : Rd(n1 +n3 +n4 ) × Rd(n2 +1) → Rd , deﬁned by e EG (xt−1 , Yt−1 ) :=

n1

A(i) yt−i +

n2

e e B (n2 −i) yt−1,t+i − yt−1,t (3.16)

i=0

i=1

+

n3 i=1

C (i) wt−i +

n4

D(i) t−i + b,

i=1

where , , . . . , yt−n , wt−1 , . . . , wt−n xt−1 = (yt−1 t−1 , . . . , t−n4 ) 1 3

and

30

3 Adaptive Learning in Linear Models e e e Yt−1 = (yt−1,t+n , . . . , yt−1,t ) . 2

Given an arbitrary state of the economy xt−1 , the error function describes all possible mean forecast errors, regardless of which forecasting rule has been applied. Geometrically, the graph of the error function (3.16) is a hyper-plane. e e , the conditional forecast error for yt−1,t vanGiven arbitrary forecasts Yt−1 ishes if and only if e e = EG (xt−1 , Yt−1 ) = 0 P − a.s. Et−1 [yt ] − yt−1,t

(3.17)

In view of Deﬁnition 2.3, Chapter 2, a forecasting rule Ψ which for given e e e ) sets Yt−1 = Ψ (xt−1 , Yt−2 ) such that (3.17) holds is called unbiased (xt−1 , Yt−2 e at (xt−1 , Yt−2 ). A forecasting rule which is unbiased along orbits can now be redeﬁned as follows. (Compare with Deﬁnition 2.4, Chapter 2.) (1)

(n +1)

Deﬁnition 3.1. A consistent forecasting rule Ψ = (Ψ , . . . , Ψ 2 ) is called unbiased if there exists a non-empty subset U ⊂ Rd(n1 +n2 +n3 +n4 +1) such that the following conditions are satisﬁed: (i) Unbiasedness. Ψ satisﬁes

EG x, Ψ (x, Y e ) = 0

for all (x, Y e ) ∈ U.

(ii) Invariance. U is invariant under the map GΨ deﬁned in (3.6), that is, for each ξ ∈ R2d , GΨ (ξ, x, Y e ) ∈ U

for all (x, Y e ) ∈ U.

Condition (i) describes states (x, Y e ) in which the forecast error (3.17) vanishes and Condition (ii) implies that (3.17) can be satisﬁed for all times t. If Ψ is unbiased in the sense of Deﬁnition 3.1, then all orbits γ(x1 , Y0e ) := e }t∈N generated by the time-one map GΨ , where initial conditions {xt , Yt−1 e (x1 , Y0 ) ∈ U may be arbitrary, correspond precisely to rational expectations equilibria (REE). 3.2.1 No-Updating Rules Observe that the error function (3.16) is globally invertible if B (0) is invertible. We can now exploit the linear structure of (3.16) and construct a linear unbiased forecasting rule as follows. Suppose B (0) is invertible. We may then solve (3.17) by setting e e yt−1,t−1+i = yt−2,t−1+i , e and solving for yt−1,t+n to get 2

i = 1, . . . , n2

3.2 Unbiased Forecasting Rules

31

(n2 +1)

e yt−1,n = Ψ 2 +1

e (xt−1 , Yt−2 ) (3.18) n n −1 1 2 e e A(i) yt−i + B (n2 −i) yt−2,t+i − yt−2,t := −(B (0) )−1 i=0 n4

i=1

+

n3

C (i) wt−i +

i=1

D(i) t−i + b ,

i=1

where xt−1 = (yt−1 , . . . , yt−n , wt−1 , . . . , wt−n , t−1 , . . . , t−n4 ) 1 3

as before.The function (3.18) induces a linear forecasting rule (1)

(n2 +1)

Ψ = (Ψ , . . . , Ψ

)

of the functional form (3.5) by setting ⎧ (i) e e e ⎨ yt−1,t−1+i = Ψ (xt−1 , Yt−2 ) := yt−2,t−1+i , ⎩ ye

(n2 +1)

t−1,t+n2

= Ψ

i = 1, . . . , n2 ,

(3.19)

e (xt−1 , Yt−2 )

e . As (3.19) never upfor an arbitrary state xt−1 and arbitrary forecasts Yt−2 dates forecasts made in the previous period, it will be referred to as a (linear) no-updating forecasting rule. By construction,

EG x, Ψ (x, Y e ) = 0 identically on Rd(n1 +n2 +n3 +n4 +1)

and since the forecasting rule (3.19) clearly fulﬁlls the consistency conditions (3.15), Condition (i) of Deﬁnition 3.1 is satisﬁed. Inserting (3.19) into the economic law (3.1), the associated time-one map (3.6) takes the form ⎧ ⎛ n1 ⎞ ⎛ ⎞ ⎛

⎞ e pr−1 0 0 ) (pr1 Yt−2 ⎪ (t + b) , 0 ⎪ ⎪ n3 ⎪ ⎠ + ⎝ (w , 0) ⎠ 0 ⎠ xt−1 + ⎝ 0 ⎨ xt = ⎝ 0 pr−1 t n4 0 0 pr−1 0 ( , 0) t ⎪ ⎪ ⎪ ⎪ ⎩ e e Yt−1 = Ψ (xt−1 , Yt−2 ) (3.20) e e with pr1 Yt−2 = yt−2,t . Since (3.20) is deﬁned globally on Rd(n1 +n2 +n3 +n4 +1) , the forecasting rule (3.19) fulﬁlls Condition (ii) of Deﬁnition 4.4, so that it is unbiased. It is easy to verify that all conditional forecast errors generated by e (3.20) vanish on average, since by construction Et−1 [yt ] = yt−1,t and hence all forecasts for yt satisfy e Et−1 [yt ] = yt−i,t

P − a.s.,

i = 1, . . . , n2 + 1,

(3.21)

as long as the no-updating rule (3.19) has been applied for the past n2 + 1 periods. Since (3.19) is deﬁned globally on Rd(n1 +n2 +n3 +n4 +1) , it is globally

32

3 Adaptive Learning in Linear Models

unbiased in the sense of Deﬁnition 3.1 implying that all orbits of (3.20) correspond to REE in the classical sense. The function (3.19) will henceforth be referred to as an unbiased no-updating forecasting rule. The forecasting rule (3.19) is constructed in such a way that Et [yt+1 ] = e e yt,t+1 for all choices of yt,t+i , i = 1, . . . , n2 . This implies a considerable freedom in selecting these ﬁrst n2 (n2 > 0) forecasts because any martingale diﬀerence sequence could, in principle, be added to these forecasts without violating the unbiasedness. This freedom constitutes a primary cause for the emergence of multiple unbiased forecasting rules and thus of REE in the presence of expectational leads. Remark 3.4. Conﬁrming Theorem 2.1 of Chapter 2, the remarkable phenomenon in economic systems with expectational leads (n2 > 0) and invertible B (0) is that the unbiased no-updating forecasting rule (3.19) generates rational expectations in the sense that all forecast errors (3.17) conditional on information available at date t vanish, including those which were made prior to that date. It yields the most precise forecasts in the sense that forecast errors vanish conditional on information which is not available at the stage in which they were issued. This fact holds only for forecasts which feed back into the system in a non-trivial manner. We note in passing that the no-updating rule (3.19) is closely related to the classical ˚ Astr¨ om-Wittenmark self-tuning regulator (˚ Astr¨ om & Wittenmark 1973), known from the literature on adaptive control and optimal tracking, cf. Caines (1988). There are well-known examples in which B (0) is not invertible, see Evans & Honkapohja (2001, Chap. 10). In these cases (3.19) does not exist and the construction of an unbiased forecasting rule may become quite involved. One way out of this problem is presented in Section 3.2.2 below. There we discuss linear minimal-state-variable predictors (MSV predictors) which generate minimalstate-variable solutions in the sense of McCallum (1983). Another way out of the problem with singular B (0) is outlined in Example 3.1. Here we exploit the fact that the ﬁrst condition in Deﬁnition 3.1 is a local invertibility property of the error function. Example 3.1. We present a particularly easy example from Wenzelburger (2006) in which B (0) is singular, indicating to what extent an unbiased forecasting rule will exist if the invertibility condition of the coeﬃcient matrix B (0) is violated. This example provides a complementary approach to the one of McCallum (1998) and may easily be extended to the general case. With the notation of Section 3.1, consider the simple economic law e + t , yt = Byt−1,t+1

(3.22)

where B ≡ B (0) is a singular d × d matrix such that the inverse of B does not exist. An example similar to (3.22) but with diﬀerent dating is found in

3.2 Unbiased Forecasting Rules

33

McCallum (1998). We may ﬁnd a ‘coordinate transformation’ z = Qy with a nonsingular d × d matrix Q, such that (12) (11) B B Q · B · Q−1 = 0 0 (11) and some d1 ×d2 matrix B (12) , where with a non-singular d1 ×d1 matrix B d1 + d2 = d. Such a coordinate transformation Q would exist, for example, if B were diagonalizable. Set zt = Qyt ,

e e zt−1,t+1 = Qyt−1,t+1 , and ηt = Qt .

(3.23)

Then (3.22) can be transformed into (e1) (1) (1) (12) (11) B zt−1,t+1 zt ηt B = , + (2) (2) (e2) 0 0 zt ηt zt−1,t+1

where e zt−1,t+1

=

(e1)

zt−1,t+1 (e2) zt−1,t+1

(3.24)

.

Thus the economic law (3.22) decouples into two (vector) equations (3.24), (2) (2) the second of which receives no feedback from the forecasts. Since zt = ηt , (e2) (2) (2) we set zt−1,t+1 = Et−1 [ηt+1 ] to obtain an unbiased forecast for zt+1 . Since (11) is invertible, we can construct an unbiased no-updating forecasting rule B for the ﬁrst component analogously to (3.19). An unbiased forecasting rule for (3.24) then is ⎧ ⎨ z (e1) (11) )−1 z (e1) − B (12) Et−1 [η (2) ] − Et−1 [ηt(1) ] t−2,t t+1 t−1,t+1 = (B . (3.25) (2) ⎩ z (e2) t−1,t+1 = Et−1 [ηt+1 ] The unbiasedness is veriﬁed by inserting (3.25) into (3.24) and taking conditional expectations, so that (1)

Et−1 [zt

(e1)

− zt−2,t ] = 0

and

(2e)

Et−2 [zt − zt−2,t ] = 0

(e1)

P − a.s. (1)

Observe that zt−2,t is a best least-squares prediction for zt

with respect to (2e)

Ft−1 and hence with respect to Ft−2 , whereas, in general, Et−1 [zt −zt−2,t ] = 0. Setting e e = Q−1 zt−1,t+1 , (3.26) yt−1,t+1 we thus obtain an unbiased forecasting rule in the sense of Deﬁnition 3.1 for the original economic law (3.22). Indeed, it follows from (3.23) that the forecast errors for (3.22) are e e = Q−1 [zt − zt−2,t ] yt − yt−2,t

P − a.s.,

34

3 Adaptive Learning in Linear Models

implying that the forecast errors conditional on Ft−2 vanish. Taking conditional expectations with respect to Ft−1 , we see in addition that even (1)

Et−1 [yt (2)

(1e)

− yt−2,t ] = 0,

(3.27)

(2)

provided that Et−1 [ηt ] − Et−2 [ηt ] = 0. The higher precision (3.27) holds, for instance, if {ηt }t∈N is a martingale diﬀerence sequence. 3.2.2 Linear MSV Predictors In the literature on rational expectations, the notion of minimal-state-variable solutions is a well-established concept to describe situations in which agents have rational expectations along particular orbits of the system. For linear models with expectational leads of order 2 this concept was developed in McCallum (1983, 1998, 1999). A univariate linear example with leads of order 3 is found in Evans & Honkapohja (2001). We relate this concept to our notion of a consistent forecasting rule by introducing minimal-state-variable predictors (MSV predictors) for linear models with expectational leads of arbitrary length. For simplicity, we assume throughout this section that n1 = n3 = 1 for the lag length of both the endogenous and exogenous variables, we let n2 > 0 be of arbitrary size, and restrict the following analysis to the white-noise case n4 = 0 with an AR(1) process {wt }t∈N and a martingale diﬀerence sequence {t }t∈N . Consider a perceived law of motion of the form ˆ t + Cw ˆ t + ˆb + t+1 , yt+1 = Ay (3.28) wt+1 = Πwt + ηt+1 , on the basis of which a forecasting agency forms expectations. yt ∈ Rd and wt ∈ Rd denote vectors of endogenous and exogenous variables, respectively, ˆ C, ˆ and Π are non-random matrices of respective dimensions, and ˆb ∈ Rd a A, non-random vector. {t }t∈N and {ηt }t∈N are martingale diﬀerence sequences on (Ω, F , P) adapted to the ﬁltration {Ft }t∈N which are assumed to be unobservable. If (yt , wt ) ∈ Rd × Rd describes the state of the economy at date t, the perceived state at date t + j derived from the model (3.28) is !j ˆ t+i−1 + ˆb + t+i ], yt+j = Aˆj yt + i=1 Aˆj−i [Cw (3.29) !j j j−i wt+j = Π wt + i=1 Π ηt+i . Taking conditional expectations with respect to Ft and using the fact that Et [wt+j ] = Π j wt , we obtain a forecasting rule Φ = (Φ(1) , . . . , Φ(n2 +1) ) by setting e yt,t+j

(j)

=Φ

(yt , wt ) := Aˆj yt +

j i=1

ˆ i−1 wt + ˆb] Aˆj−i [CΠ

(3.30)

3.2 Unbiased Forecasting Rules

35

for all j = 1, . . . , n2 + 1. Since the forecasting rule Φ deﬁned in (3.30) uses a minimal amount of endogenous and exogenous variables, it will be referred to as minimal-state-variable predictor (MSV predictor). By construction, one e = Et [yt+j ] if yt+j is thought of being generated from (3.29), and a has yt,t+j straightforward calculation shows that each forecast (3.30) satisﬁes e e ] = Et−1 [Φ(j) (yt , wt )] = Φ(j+1) (wt−1 , yt−1 ) = yt−1,t+j Et−1 [yt,t+j

for all j = 1, . . . , n2 . This implies that a MSV predictor is a consistent forecasting rule, since it satisﬁes Condition (3.15). Observe that the MSV predictor ˆ C, ˆ ˆb, Π). associated with (3.28) can be represented by the quadruple (A, In order to construct a MSV predictor Φ = (Φ(1) , . . . , Φ(n2 +1) ) which is unbiased in the sense of Deﬁnition 3.1, let " # ΓΦ := (y, w, z (1) , . . . , z (n2 +1) ) ∈ Rd(n2 +3) z (j) = Φ(j) (y, w), 1 ≤ j ≤ n2 + 1 be the graph of the MSV predictor Φ = (Φ(1) , . . . , Φ(n2 +1) ). It is easy to see that ΓΦ is invariant under the corresponding time-one map GΦ . Hence Φ satisﬁes Property (ii) of Deﬁnition 3.1. A MSV predictor Φ is unbiased in the sense of Deﬁnition 3.1 if

(3.31) EG y, w, Φ(1) (y, w), . . . , Φ(n2 +1) (y, w) = 0 for all (y, w) ∈ Rd × Rd , where EG denotes the error function associated with the economic law (3.1) (with n1 = n3 = 1 and n4 = 0). Inserting (3.30) into (3.31), a MSV predictor Φ is unbiased if and only if ˆ and ˆb ∈ Rd in (3.30) satisfy the unknown d × d coeﬃcient matrices Aˆ and C, 0 = A(1) + [B (n2 ) − Id ]Aˆ +

n 2 −1

B (n2 −i) Aˆi+1 ,

(3.32)

i=0

0=

n i+1 2 −1

ˆ j−1 + [B (n2 ) − Id ]Cˆ + C (1) , B (n2 −i) Aˆi+1−j CΠ

i=0 j=1

⎞

⎛

n i+1 2 −1

0=⎝

(3.33)

B (n2 −i) Aˆi+1−j + [B (n2 ) − Id ]⎠ ˆb + b.

(3.34)

i=0 j=1

Summarizing, this yields the following existence theorem for unbiased MSV predictors. Theorem 3.1. Suppose that n1 = n3 = 1, n4 = 0, and that the exogenous observable noise {wt }t∈N is an AR(1) process of the form wt = Πwt−1 + ηt ,

36

3 Adaptive Learning in Linear Models

where Π is a non-random d × d matrix whose eigenvalues lie within the unit circle and {ηt }∈N is a martingale diﬀerence sequence. Let (1)

(n2 +1)

Φ = (Φ , . . . , Φ

)

be a MSV predictor pertaining to the perceived law of motion (3.28) which is represented by the quadruple (Aˆ , Cˆ , ˆb , Π), where Aˆ and Cˆ are d × d matrices with real coeﬃcients and ˆb ∈ Rd . Then Φ is unbiased for the economic law (3.1) in the sense of Deﬁnition 3.1 if and only if the triplet (Aˆ , Cˆ , ˆb ) is a solution of eqs. (3.32)-(3.34). Eqs. (3.32)-(3.34) correspond to Eqs. (10.5)-(10.7) in Evans & Honkapohja (2001, p. 230) for expectational leads of arbitrary length. It is shown in Lancaster (1969, Thm. 8.4.1, p. 262) that (3.33) can be transformed into a linear matrix equation. Thus (3.33) and (3.34) are linear equations for Cˆ and ˆb, once Aˆ is known. It is straightforward to verify that Aˆ is a solution to (3.32) if and only if the linear factor (λId − Aˆ ), λ ∈ R is a right divisor of the matrix polynomial L(λ) = A(1) + [B (n2 ) − Id ]λ +

n 2 −1

B (n2 −i) λi+1 ,

λ ∈ R,

i=0

see Gohberg, Lancaster & Rodman (1982, p. 125). Solutions Aˆ with real coeﬃcients need neither exist nor be uniquely determined because, in general, there are multiple right divisors. In fact, the full set of solutions may be obtained from a decomposition of L(λ) into linear factors, cf. Gohberg, Lancaster & Rodman (1982, p. 113). If Φ is an unbiased MSV predictor with coeﬃcients, it follows from Theorem 3.1 together with (3.31) that

(1) yt = Φ (yt−1 , wt−1 ) + t = G ξt , yt−1 , wt−1 , Φ (yt−1 , wt−1 )

(3.35)

for all t ∈ N, where G denotes the economic law with n1 = n3 = 1, n4 = 0 as deﬁned in (3.4). In other words, unbiased MSV predictors, if they exist, generate minimal-state-variable solutions to (3.1) in the sense of McCallum (1983). These correspond to the perceived law of motion (3.28) with coeﬃcients (Aˆ , Cˆ , ˆb , Π). All forecasts obtained from Φ are best least-squares predictions with respect to the corresponding information along any orbit of the resulting time-one map (3.6). To compute an unbiased MSV predictor may become a diﬃcult problem. Explicit solutions to (3.32)-(3.34) for the case n2 = 1 are found in McCallum (1983, 1998, 1999). The existence of solutions to (3.32)-(3.34), in general, do not require an invertible B (0) . However, it does requires full knowledge of the coeﬃcient matrix Π which is not needed for a no-updating rule. Notice also that all eigenvalues of a solution Aˆ must have modulus less than unity in

3.3 Approximating Unbiased Forecasting Rules

37

order to obtain a stable process and thus an economically meaningful solution. In Example 2.2 of Chapter 2 we have already seen that unbiased MSV predictors may not exist, while unbiased no-updating rules exist. Despite this fact, however, unbiased MSV predictors are important as they may generate stable long-run behavior under rational expectations in an otherwise unstable situation. Remark 3.5. In some cases linear models may admit unbiased MSV predictors which are non-linear functions, cf. Froot & Obstfeld (1991). These correspond to non-linear rational-expectations solutions and will be treated in Chapter 4.

3.3 Approximating Unbiased Forecasting Rules We are now in a position to introduce a learning scheme for unbiased forecasting rules for the general linear system (3.1) using methods from control theory which are developed in Lai & Wei (1986b). For simplicity of exposition, we assume that the constant term in (3.1) is zero, i.e. b = 0. Setting θ = (A(1) , . . . , A(n1 ) , B (0) , . . . , B (n2 ) , C (1) , . . . , C (n3 ) , D(1) , . . . , D(n4 ) ) and

e Xt−1 = (Yt−1 , Yt−1 , Wt−1 , Et−1 ) ,

(3.36)

the linear system (3.1) may be rewritten as yt = θ Xt−1 + t .

(3.37)

We will use the representation (3.37) to approximate the unbiased no-updating rule Φ given in (3.18) from estimates for the coeﬃcients θ in two steps. In the ﬁrst step, we estimate the unknown coeﬃcient matrix θ from historical data. In the second step we compute an approximation of an unbiased forecasting rule. The estimates of the unknown coeﬃcient matrix θ in (3.37) are obtained from the so-called approximate-maximum-likelihood (AML) algorithm, cf. Lai & Wei (1986b). This method provides a recursive scheme which generates successive estimates (1) (n ) ˆ (0) ˆ (n2 ) , Cˆ (1) , . . . , Cˆ (n3 ) , D ˆ (1) , . . . , D ˆ (n4 ) ) θˆt = (Aˆt , . . . , Aˆt 1 , B t , . . . , Bt t t t t (3.38) for θ based on information available at date t. Since the noise process {t }t∈N is assumed to be unobservable, we replace the regressor Xt−1 in (3.37) by

ˆ t−1 := (Y , Y e , W , E ˆ ) , X t−1 t−1 t−1 t−1 ˆt−1 := (ˆ E ˆ t−1 , . . . , t−n4 ) ,

where

(3.39)

38

3 Adaptive Learning in Linear Models

ˆ t−i−1 , ˆt−i := yt−i − θˆt−i X

i = 1, . . . , n4

(3.40)

are also called a-posteriori prediction errors. The AML algorithm is recursively deﬁned by ⎧ ˆ t−1 )X ˆ Pt−1 , ⎨ θˆt = θˆt−1 + (yt − θˆt−1 X t−1 (3.41) ⎩P −1 ˆ ˆ ˆ ˆ , t−1 = Pt−2 − Pt−2 (Xt−1 Xt−1 )Pt−2 (1 + Xt−1 Pt−2 Xt−1 ) where the initial conditions P0 and θˆ0 may be chosen arbitrarily with P0 invertible.1 The parameter estimate θˆt as given in (3.38) can now be used to estimate an unbiased no-updating rule (3.19) as follows. Let B # denote the generalized inverse of a matrix B, also referred to as Moore-Penrose inverse, see Lancaster (1969, p. 303). B # coincides with the inverse B −1 if B is invertible. An approximation of the unbiased no-updating rule (3.19) is given by the map e yt,t+j

e = yt−1,t+j ,

j = 1, . . . , n2 ,

e ˆ xt , Y e ; θˆt ) yt,t+1+n = ψ(ˆ t−1 2 (0)#

ˆt := −B

n1

(i) Aˆt yt+1−i +

e ˆt(n2 −i) yt−1,t+1+i B

n3

(i) Cˆt wt+1−i +

i=1

where

(3.42)

i=0

i=1 e −yt−1,t+1 +

n 2 −1

n4

ˆ (i) ˆt+1−i , D t

i=1

ˆt ) x ˆt := (Yt , Wt , E

is an approximation of the vector of endogenous variables xt deﬁned in (3.2). Using (3.41), the forecasting rule (3.42) will then be updated over time with the updates of the parameter estimates θˆt . Remark 3.6. (i) By Theorem 2 of Lancaster (1969, p. 306), x0 := B # b minimizes Bx − b in the sense that any x = x0 ∈ Rd satisﬁes either

Bx − b > Bx0 − b or Bx − b = Bx0 − b and x > x0 . (ii) If B (0) is known to be invertible, then an approximation of the unbiased no-updating rule could directly be obtained from regressing over (3.18) instead of (3.1), thereby avoiding the somewhat involved computation of a Moore-Penrose inverse. (iii) Instead of (3.42), an approximation of any other unbiased forecasting rule could be taken. Observe, however, that taking an MSV predictor as introduced in Section 3.2.2 may be computationally demanding. Eqs. (3.32)(3.34) may not admit solutions with real coeﬃcients for approximated parameters. With the focus on global convergence of the AML algorithm, we therefore choose (3.42). 1

−1 ˆ t , cf. Caines (1988). ˆt X By the matrix inversion lemma, Pt−1 = Pt−1 +X

3.4 Convergence of an AML-Based Learning Scheme

39

(iv) A constant term b = 0 could be incorporated into the as

AML !n4algorithm follows. The simplest way is to assume that ¯ with Id + i=1 D(i) ¯ = b exists. We may then set ˜t := t + ¯ and replace t with ˜t in the deﬁnition 1 !t ˆi is a of the a-posteriori prediction errors (3.40). Then ˆbt := t+1 i=0 period-t estimate for b. By a reasoning analogous to the proof of Theorem 3.3 below, ˆt will converge P-a.s. to t in either case, so that ˆbt converges to b.

3.4 Convergence of an AML-Based Learning Scheme We will now investigate under which conditions a learning scheme based on the AML algorithm (3.41) and the forecasting rule (3.42) converges to rational expectations. Under a number of assumptions discussed in this section, global convergence obtains if the coeﬃcient estimates θˆt generated by the AML algorithm are consistent in the sense that θˆt converges a.s. to θ, independently of the initial state of the system and independently of initial parameter estimates. Lai & Wei (1986b) show that under certain standard assumptions (see Proposition 3.3 below) θˆt converges to θ a.s. if the regressors Xt given in (3.7) satisfy the weak persistent excitation condition $! % T λmin t=1 Xt Xt $ $! %% → ∞ P − a.s. as T → ∞, (3.43) T log e + λmax t=1 Xt Xt where λmin (B) and λmax (B) denote the minimum and maximum eigenvalues of a symmetric matrix B, respectively. It has long been recognized that in feedback systems of the form (3.1), (3.43) may be violated such that the parameter estimates may fail to be consistent. The main problem with (3.43) is that the regressors Xt cannot be manipulated directly. However, Lai & Wei (1986b) showed that the weak persistent excitation condition (3.43) can be translated into conditions on the inputs alone which in our case are the forecasts and the two exogenous variables. Following Lai & Wei (1986b), Condition (3.43) will now be replaced by more direct assumptions concerning these inputs alone. This requires a modiﬁcation of the forecasting rule (3.42). The corresponding assumptions are stipulated as follows. Assumption 3.1. The unobservable noise process {t }t∈N is a {Ft }t∈N -adapted martingale diﬀerence sequence on the probability space (Ω, F , P) with values in Rd which satisﬁes the following conditions. (i) supt≥1 E t α Ft−1 < ∞ P − a.s. for some α > 2. !T 1 (ii) T = o(T 2 ), t=1 t 2 = O(T ), and % $! T lim inf T1 λmin > 0 P − a.s. t t t=1 T →∞

40

3 Adaptive Learning in Linear Models

(iii) Let D(0) , . . . , D(n4 ) , n4 ≥ 0, be non-random d × d matrices, where D(0) = Id is the d × d identity matrix. Assume that the matrix polynomial Γ (s) = D(0) + D(1) s + · · · + D(n4 ) sn4 ,

s∈C

is strictly positive real, i.e., (a) det Γ (s) = 0 for all s ∈ C with |s| ≤ 1, and s), with s¯ ∈ C (b) for each s ∈ C with |s| ≤ 1, the matrix Γ (s) + Γ (¯ denoting the complex conjugate of s, is strictly positive deﬁnite, that is,

ζ Γ (s) + Γ (¯ s) ζ > 0 for all 0 = ζ ∈ Rd .

Assumption 3.2. The observable exogenous variables {wt }t∈N are given by a {Ft }t∈N -adapted stochastic process on the probability space (Ω, F , P) with values in Rd . Let Ft := σ(Ft−1 , t ) denote the σ-algebra generated by Ft−1 and t , where t is deﬁned in Assumption 3.1. Setting w ˜t = wt − E[wt |Ft ], the process {wt }t∈N is assumed to satisfy the following conditions. (i) sup E[ w ˜t β |Ft ] < ∞

P − a.s. for some β > 2.

t≥1

% w ˜ w ˜ > 0 P − a.s. t t t=1 $ %% $ !T !T 1 P − a.s. ˜t w ˜t (iii) wT = o(T 2 ) and t=1 E[wt |Ft ] 2 = O λmin t=1 w (ii)

!T

˜t 2 = O(T ) and lim inf t=1 w

1 T →∞ T

λmin

$! T

The process deﬁned in Assumption 3.1 is referred to as colored noise. The white-noise case in which Γ (s) = Id and n4 = 0 assumed in most economic models satisﬁes Assumption 3.1. The case in which the observable exogenous process {wt }t∈N is generated by a stable AR(1) process wt = Πwt−1 + ηt , where Π is a non-random d × d matrix whose eigenvalues lie within the unit circle and {ηt }t∈N is white noise and hence stochastically independent of the unobservable noise {t }t∈N , is an example for a process that fulﬁlls Assumption 3.2. Assumptions 3.1 and 3.2 are thus generalizations of popular parameterizations in the learning literature, cf. Evans & Honkapohja (2001). Remark 3.7. (i) Since Ljung (1977) Assumption 3.1 appears to be fundamental in the systems and control literature, cf. Caines (1988). Ljung, S¨ oderstr¨ om & Gustavsson (1975) provide counter-examples to the strong consistency of the AML algorithm if the so-called positive real condition, i.e., Assumption 3.1 (iii) is violated. (ii) Assumption 3.1 contains conditions which are redundant. Condition (i) implies ∞ % $ P t α > δtαα | Ft−1 < ∞ t=1

3.4 Convergence of an AML-Based Learning Scheme

41

for every α > α1 and each δ > 0. By the conditional Borel-Cantelli Lemma (Stout 1974, Thm. 2.8.5, p. 55), T = o(T α ) P-a.s. for each α > α1 and thus in particular for α = 12 . It is shown in Lai & Wei (1983) that Condition (ii) follows from Condition (i) and the weaker condition (ii )

lim inf λmin (E[t t |Ft−1 ]) > 0 t→∞

P − a.s.

Analogous remarks apply to Assumption 3.2 and to Assumption 3.3 below. Our last assumption concerns the controllable inputs which are the forecasts. The key idea is that the forecasts are censored whenever the persistent excitation condition (3.43) is violated or the forecasts itself become unbounded. Assumption 3.3. A forecasting agency issues forecasts based on the AML algorithm (3.41) and the forecasting rule ψˆ deﬁned in (3.42) as follows. Given two sequences of real numbers {ct }t∈N and {vt }t∈N , the forecasts in period t are based on the current parameter estimate θˆt and given by a map (1) (n +1) Ψˆt = (Ψˆt , . . . , Ψˆt 2 ),

(3.44)

deﬁned by e yt,t+j e yt,t+1+n 2

(j) e e = Ψˆt (ˆ xt , Yt−1 ; θˆt ) := yt−1,t+j , j = 1, . . . , n2 , (n2 +1) e ˆ ˆ = Ψt (ˆ xt , Yt−1 ; θt ) ⎧ ˆ xt , Y e ; θˆt ) + vt ˆ xt , Y e ; θˆt ) ≤ ct , ⎪ if ψ(ˆ ⎨ψ(ˆ t−1 t−1 c := t e ˆ ˆ ψ(ˆ xt , Yt−1 ; θt ) + vt otherwise. ⎪ ⎩ ψ(ˆ ˆ xt , Y e , θˆt )

t−1

Here for a given tolerance level ctol > 0, the bounds {ct }t∈N consists of a 1 sequence of positive numbers satisfying ct ≥ ctol for all t ∈ N, ct = o(t 2 ), !T 2 and t=1 ct = O(T ). The dither {vt }t∈N is an exogenous white-noise process satisfying the following conditions. (a) For each t ∈ N, let Gt := σ(Ft−1 , yt , wt ) denote the σ-algebra generated by Ft−1 , yt , and wt . Assume E[vt |Gt ] = 0 for all t ∈ N and supt≥1 E[ vt γ |Gt ] < ∞ P-a.s. for some γ > 2, and (b) lim inf λmin (E[vt vt |Gt ]) > 0 P-a.s. t→∞

In view of Corollary 3.1, the sequence {ct }t∈N ceils the growth of the forecasts and in this sense censors the forecasts as to ensure stability. The dither vt in (3.44) may be thought of being deliberately applied by a forecasting agency in order to satisfy the excitation condition (3.43). It will turn out Proposition

42

3 Adaptive Learning in Linear Models

3.3 below that this guarantees convergence of the AML algorithm. Observe that no requirement on the magnitude of the dither is made in Assumption 3.3 so that the size of this perturbation may be very small. The series of forecasts generated by the AML algorithm (3.41) and the forecasting rule (3.44) deﬁne an AML-based adaptive learning scheme for the linear system (3.1). With the notation of Section 3.1, the system under AMLbased adaptive learning evolves over time according to the set of equations ⎧ˆ θt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Pt−1 ˆt E ⎪ ⎪ ⎪ ⎪ Ye ⎪ ⎪ ⎩ t xt+1

ˆ t−1 )X ˆ Pt−1 , = θˆt−1 + (yt − θˆt−1 X t−1 ˆ )Pt−2 (1 + X ˆ t−1 )−1 , ˆ t−1 X ˆ Pt−2 X = Pt−2 − Pt−2 (X t−1 t−1

ˆ t−1 ) , (prn4 E ˆ = (yt − θˆt X , −1 t−1 ) ˆt , Y e ; θˆt ), = Ψˆt (Yt , Wt , E

(3.45)

t−1

= G(ξt+1 , xt , Yte ).

The last condition required for consistent estimates is an identiﬁability condition. Let A(s), B(s), C(s), and D(s), s ∈ C, be matrix polynomials of the form A(s) = Id sn1 − A(1) sn1 −1 − · · · − A(n1 ) , B(s) = B (0) sn2 + B (1) sn2 −1 + · · · + B (n2 ) , (3.46) C(s) = C (1) sn3 −1 + · · · + C (n3 ) , D(s) = Id sn4 + D(1) sn4 −1 + · · · + D(n4 ) , where Id denotes the d × d identity matrix and A(i) , B (i) , C (i) , and D(i) are the d × d parameter matrices of the economic law (3.1). The matrix polynomials A(s), B(s), C(s), and D(s) are said to be left coprime, if their greatest common left divisors are unimodular (i.e., with constant determinants = 0). Equivalently, A(s), B(s), C(s) and D(s) are left coprime, if there exist matrix polynomials K(s), L(s), M (s), and N (s) such that A(s)K(s) + B(s)L(s) + C(s)M (s) + D(s)N (s) = Id , cf. Kailath (1980, p. 399) or Hannan & Deistler (1988, Chap. 2). The left-coprime assumption is often referred to as an identiﬁability assumption for an ARMAX system with colored noise. It is automatically satisﬁed for the white-noise case. We are now ready to state our main theorems. Theorem 3.2. Consider the system (3.1). Let the following hypotheses be satisﬁed: (i) All eigenvalues λi of the matrix A deﬁned in (3.3) lie on or within the unit circle, that is, |λi | ≤ 1. (ii) Identiﬁability assumption. The matrix polynomials A(s), B(s), C(s), and D(s) given in (3.46) are left coprime. (iii) The exogenous perturbations satisfy Assumptions 3.1 and 3.2. (iv) Forecasts are generated by the AML-based learning scheme as deﬁned in Assumption 3.3.

3.4 Convergence of an AML-Based Learning Scheme

43

Then the system (3.45) is stable in the sense of Proposition 3.1 and the AMLbased adaptive learning scheme generates strongly consistent estimates, i.e., θˆt → θ P-a.s. as t → ∞, globally for all initial conditions. The proof of Theorem 3.2 is quite involved and will be carried out in Section 3.5.2 below. Conditions (i), (iii), and (iv) of Theorem 3.2 will ensure that the system (3.45) is stable in the sense of Proposition 3.1. Condition (ii) is needed to ensure strongly consistent parameter estimates. It should be emphasized that only the dither and the bounds for the forecasts in Assumption (iv) matter, but not the particular functional form of the forecasting rule. Theorem 3.2 makes no assumption on the stability of the system (3.1) under rational expectations. So by the following Theorem 3.3, a forecasting agency will apply the approximated unbiased forecasting rule only if the system is stable under that forecasting rule. Theorem 3.3. Let the hypotheses of Theorem 3.2 be satisﬁed and assume, in addition, that the following holds: (i) All eigenvalues λi of the matrix A deﬁned in (3.3) lie within the unit circle, i.e., |λi | < 1. (ii) The system (3.1) under the unbiased no-updating rule (3.19) is stable in the sense that all eigenvalues of the coeﬃcient matrix of the system (3.20) lie within the unit circle. (iii) The parameter estimates generated by (3.20) converge fast enough so that ˆ t−1 → 0

θ − θˆt X

P − a.s. as t → ∞.

Then the tolerance level ctol may be chosen large enough so that all orbits of the AML-based learning scheme converge to orbits with rational expectations, t − t → 0 P-a.s. as t → ∞. globally on Rd(n1 +n2 +n3 +n4 +1) . Moreover, ˆ Observe that Condition (iii) is automatically satisﬁed if all exogenous noise ˆ t }t∈N is bounded. This observation will is bounded, because in this case {X become clear in below’s proof of Theorem 3.3. Corollary 3.3 below provides convergence rates for the AML algorithm in terms of the observable design matrices Pt deﬁned in (3.41), which are

−1 P log λ max

t P − a.s.,

θˆt − θ 2 = O λmin Pt−1 −1 ˆ . These bounds indicate how precise the actual ˆtX +X wherePt−1 = Pt−1 t parameter estimates are, so that the dither could be removed as soon as a certain threshold is reached. Thus, the AML-based learning scheme allows a forecasting agency to select an approximation of the desired forecasting rule from actual parameter estimates at any point in time. In particular, the noupdating rule used in Assumption 3.3 could be replaced by any forecasting

44

3 Adaptive Learning in Linear Models

rule which generates minimal-state-variable solutions provided this rule exists. As soon as parameter estimates are suﬃciently close to their true values, this enables an agency to select a stable minimal-state-variable solution if it exists. Remark 3.8. (i) The convergence of the AML algorithm has ﬁrst been investigated by Ljung (1977). His stability analysis is based on a limiting non-random ordinary diﬀerential equation. We follow a martingale approach initiated by Solo (1979) and later extended by Lai & Wei (1986b). The weak persistent excitation condition is considerably less restrictive than the assumptions used by Ljung (1977) and Solo (1979). See Lai & Wei (1986b, pp. 228-231) for an account of the literature. (ii) The AML algorithm is an extension of the ordinary-least-squares (OLS) algorithm as usually applied in the learning literature to a class of more general noise processes. It is also referred to as recursive extended-leastsquares (RELS) algorithm (Caines 1988, p. 540 and p. 557) or recursive prediction-error (RPE) algorithm (Ljung & S¨ oderstr¨ om 1983, Chap. 3.7). (iii) The concept of continually disturbed inputs, referred to as dither, is originally due to Caines, e.g., see Lai & Wei (1986b, p. 247). It is generalized in Lai & Wei (1986a, 1987) to the concept of occasional excitation of the inputs, where the dither is exerted only if the excitation condition is violated. For an application of these techniques, we refer to their work. (iv) Section 3.2.2 showed that contrary to (3.42), MSV predictors require full knowledge of the process that governs the exogenous observables. Insufﬁcient estimates for that process may result in a failure of the learning scheme. For example, if the exogenous process {wt }t∈N is an AR(1) process, then the design matrix pertaining to the AML estimates of the corresponding coeﬃcients may violate the persistent excitation condition (3.43). As a consequence, strongly consistent estimates for these coeﬃcients may be unavailable, cf. Lai & Wei (1986a).

Summary and Conclusions We introduced a learning scheme based on the AML algorithm which converges globally for all initial conditions under standard assumptions. Although the scheme can be used for linear models only, the analysis reveals that it is advantageous to ﬁrst estimate the whole system including the feedback of any forecast acting on the system itself and then compute an approximation of an unbiased forecasting rule which generates the desired rational expectations equilibria. Resolving an issue put forward in McCallum (1999), this approach allows to select an approximation of a preferred forecasting rule on the basis of an adequate economic reasoning, as soon as parameter estimates are sufﬁciently precise. The mathematical object to be estimated from time series data was the error function associated with the economic law. Methodologically, we distinguished between four separate issues. First, the existence of a desired unbiased forecasting rule; second, the dynamic stability

3.5 Mathematical Appendix

45

of the system under an unbiased forecasting rule; third, the dynamic stability of the system under the applied learning scheme and, ﬁnally, the success of the learning scheme in terms of providing strongly consistent estimates of the system’s parameters.

3.5 Mathematical Appendix 3.5.1 On the Stability of Linear Systems We begin with the proofs of Proposition 3.1 and Corollary 3.1. Proof of Proposition 3.1. The proof is an adaptation of Lai & Wei (1982, Thm. 2 (i), p. 161). Let n2 n3 n4

e B (n2 −i) yt−1,t+i + C (i) wt−i + t + D(i) t−i + b) , 0, . . . , 0 . Zt = ( i=0

i=1

i=1

Using (3.3) and the notation introduced above, the system (3.4) may be rewritten as Yt = AYt−1 + Zt and thus YT = AT Y0 +

T

AT −t Zt .

(3.47)

t=1

Let λ denote the maximum modulus of the eigenvalues of A and M = maxj mj with mj denoting the multiplicity of the j-th eigenvalue of A. By Varga (1962, Thm. 3.1, p. 65), there exists a positive constant c > 0 such that2

AT ∼ cT M−1 λ

T

as T → ∞.

(3.48)

Since by assumption λ ≤ 1, AT = O(T M−1 ). In view of Assumptions (ii) and (3.8), we have Zt = o(tβ ) and therefore by (3.47) and (3.48), there exists some α > 0 so that

YT ≤ AT

Y0 +

T

AT −t

Zt ≤ O(T α )

P − a.s.

t=1

This proves the assertion.

Proof of Corollary 3.1. In view of the assumptions, Zt = o(tβ ). Since λ < 1, (3.48) implies Aτ ≤ µ < 1 for some τ > 0 so that 2 By at ∼ bt P-a.s. for two sequences of random numbers {at } and {bt }, we mean that lim abtt = 1 P-a.s. t→∞

46

3 Adaptive Learning in Linear Models

YT ≤ AT

Y0 +

T

AT −t

Zt = o(T β )

P − a.s.

t=1

For sake of completeness, we state the following corollary. Corollary 3.2. Consider the linear system, t ∈ N.

Yt = AYt−1 + Zt ,

Assume that all eigenvalues of A lie within the unit circle and that T

Zt 2 = O(T ).

t=1

Then

T

Yt 2 = O(T ).

t=1

Proof. Note ﬁrst that the Schwarz inequality implies t T t=1

2

A

t−s

Zs

≤

s=1

t T t=1

≤

∞

A

t−s

s=1

A

t

t=0

≤

∞

t

A

t−s

s=1

T T

At

t=0

T

A

t−s

Zs 2

t=s

s=1

2

Zs

2

Zs 2 .

s=1

In view of (3.47), it follows that T t=1

Yt ≤ 2 2

T

A

Y0 +2

t 2

2

t=1

∞

2

A

t=0

t

T

Zs 2 = O(T )

P−a.s.

s=1

The following approximation result will be needed in the convergence proof of Theorem 3.3 below. Proposition 3.2. Consider two linear systems, one given by Yt = AYt−1 + Zt , and another one deﬁned by

t ∈ N,

3.5 Mathematical Appendix

ˆ t Yˆt−1 + Zˆt , Yˆt = A

47

t ∈ N,

ˆ t }t∈N is a sequence of random matriwhere A is a non-random matrix, {A ˆ ces, and {Zt }t∈N and {Zt }t∈N are two sequences of random vectors which are bounded P-a.s. Let all eigenvalues of A lie within the unit circle and assume that P-a.s. ˆT = 0 lim A − A

and

T →∞

lim ZT − ZˆT = 0

T →∞

Then the sequence lim YT − YˆT is bounded P-a.s. T →∞

Proof. For arbitrary t ∈ N, we have Yt+j = Aj Yt +

j

Aj−i Zt+i

i=1

and Yˆt+j =

j &

ˆ t+i Yˆt + A

i=1

j j−1 $ &

% ˆ t+k Zˆt+i + Zt+j . A

i=1 k=i+1

Thus

Yt+j − Yˆt+j ≤ Aj

Yt − Yˆt +

j

Aj−i

Zt+i − Zˆt+i

i=1 j j−1 ' j ' ' ' & & ' ' j−i ˆ t+i ' ˆ t+k ' +'Aj − − A A ' Yˆt + 'A ' Zˆt+i . i=1

i=1

k=i+1

For each δ > 0 there exists t ∈ N such that P-a.s. ˆ t+i < δ

A − A

and Zt+i − Zˆt+i < δ for all i ≥ 1

The proposition then follows from the eigenvalue condition and the bounded ness of {Zˆt }t∈N . 3.5.2 Proofs of Theorem 3.2 and Theorem 3.3 The proof of Theorem 3.2 requires a series of Propositions and Theorems which will be developed next. Following Lai & Wei (1986b), we ﬁrst provide a general theorem which establishes strong consistency of the AML algorithm if the weak persistent excitation condition holds. To this end we denote by λmin (B) and λmax (B) the minimum and maximum eigenvalues of a symmetric matrix B, respectively.

48

3 Adaptive Learning in Linear Models

Proposition 3.3. (Lai & Wei 1986b, Thm. 4, p. 241) Consider the stochastic regression model yt = Aηt−1 + t + D(1) t−1 + · · · + D(n4 ) t−n4 , where A is a d × k non-random matrix, D(1) , . . . , D(n4 ) are non-random d × d matrices, and {t }t∈N is a stochastic process which satisﬁes Assumption 3.1. Let ηt be Ft measurable and θ = (A, D(1) , . . . , D(n4 ) )

and

Xt−1 = (ηt−1 , t−1 , . . . , t−n4 ) .

Consider the AML algorithm, given by ⎧ ˆ t−1 )X ˆ t−1 ⎨ θˆt = θˆt−1 + (yt − θˆt−1 X Pt−1 , ⎩ P −1 = P −1 + X ˆ t−1 ˆ t−1 X , t−1 t−2 where ˆ t−1 := (η , ˆ , . . . , ˆ X t−1 t−1 t−n4 )

If the weak persistent excitation condition $! % T X X λmin t t t=1 $! %% → ∞ $ T log e + λmax X X t=1 t t

and

ˆ t−1 . ˆt := yt − θˆt X

P − a.s. as T → ∞

(3.49)

holds, then θˆt → θ

P − a.s. as t → ∞.

Corollary 3.3. Under the hypotheses of Proposition 3.3,

log λmax Pt−1 2 ˆ

P − a.s.

θt − θ = O λmin Pt−1 The proof of Corollary 3.3 is found in Lai & Wei (1986a). Let A(s) denote the adjoint of the matrix A(s) such that A(s)A(s) = a(s)Id with a(s) = !dn1 j det A(s). We have a(s) = j=0 aj s for suitable real coeﬃcients aj with a0 = 1. Lemma 3.1. Let A(s), K1 (s), . . . , Km (s) be left coprime matrix polynomials. Assume that the degree of the matrix polynomial A(s) is deg A(s) = n1 with ˜ the identity matrix as leading coeﬃcient and let A(s) denote the adjoint of A(s) with a(s) = det A(s). If there exist matrix polynomials N1 (s), . . . , Nm (s) and vectors α1 , . . . , αn1 ∈ Rd such that n1

i−1 ˜ α = a(s)Nj (s) i A(s)Kj (s)s

i=1

then α1 = · · · = αn1 = 0.

for all j = 1, . . . , m,

(3.50)

3.5 Mathematical Appendix

49

Proof. By the coprimeness, there exist matrix polynomials M0 (s), . . . , Mm (s) such that A(s)M0 (s) + K1 (s)M1 (s) + · · · + Km (s)Mm (s) = Id . Suppose that (3.50) holds. Then n1

i−1 α i A(s)s

=

n1 i=1

i=1

α i A(s) ⎡

= a(s) ⎣

n1 i=1

A(s)M0 (s) +

α i M0 (s) +

m

m

Kj (s)Mj (s) si−1

j=1

⎤

Nj (s)Mj (s)⎦ .

(3.51)

j=1

Since the degree of A(s) is deg A(s) = (d − 1)n1 and the degree of a(s) is deg a(s) = dn1 , n 1 i−1 = dn1 − 1 ≤ deg a(s). deg αi A(s)s ≤ n1 − 1 + deg A(s) i=1

!n1 i−1 ˜ A(s) = 0 for all s. Since the leading It follows from (3.51) that i=1 αi s ˜ coeﬃcient of A(s) ! is the identity matrix, A(s) and A(s) have full rank for n1 i−1 = 0 for almost all s and therefore α1 = · · · = almost all s. Thus i=1 αi s αn1 = 0.

Lemma 3.2. Let A and be an d1 × d2 matrix with kerA = {0} and B be a symmetric d2 × d2 matrix. Then there exists ρ > 0 with λmin (ABA ) ≥ ρλmin (B). Proof. By Theorem 3.2.1, Lancaster (1969, p. 109), any symmetric matrix C satisﬁes λmin (C) = minx=1 (x Cx). Since AA is nonsingular, for any 0 = x ∈ Rd1 , x ABA x x ABA x x AA x x AA x = · ≥ λmin (B). x x x AA x x x x x The lemma follows from another application of Theorem 3.2.1 with ρ = λmin (AA ) > 0.

Having stated these basic results, we need the following representation of the system (3.1). Letting ι denote the unit shift operator (i.e., ιXt = Xt+1 ), the system (3.1) may be rewritten as A(ι)yt−n1 = B(ι)ut−n2 +1 + C(ι)wt−n3 + D(ι)t−n4 ,

(3.52)

e for the forecasts. The next Proposition where we have set ut−1 = yt−1,t+n 2 3.4 is a version of Theorem 2 in Lai & Wei (1986b, p. 235) for an additional source of inputs.

50

3 Adaptive Learning in Linear Models

Proposition 3.4. Suppose that the matrix polynomials A(s), B(s), C(s), and D(s) given (3.46) are left coprime and that (3.52) holds for all times t. Let = (yt−1 , . . . , yt−n , u Xt−1 t−1 , . . . , ut−n2 −1 , wt−1 , . . . , wt−n3 , t−1 , . . . , t−n4 ), 1 = (u zt−1 t−1+dn1 , . . . , ut−n2 +1 , wt−1+dn1 , . . . , wt−n3 , t−1+dn1 , . . . , t−n4 ). (3.53) Then there exists ρ > 0 such that for all T ≥ dn1 , T T ≥ ρλmin Xt Xt zt zt . λmin t=1

t=1

Proof. Let A(s) denote the adjoint of the matrix A(s) as before. We have !dn1 a(s) = j=0 aj sj , (a0 = 1) and thus a(ι)Xt−1 =

dn1

aj Xt−1+dn1 −j .

j=0

Since A(ι)A(ι)X t−1 = a(ι)Xt−1 , (3.52) implies

a(ι)yt−j = A(ι) B(ι)ιn1 −j ut−n2 +1 + D(ι)ιn1 −j wt−n3 + C(ι)ιn1 −j t−n4 , a(ι)ut−j = a(ι)ιn2 +1−j ut−n2 +1 , a(ι)wt−j = a(ι)ιn4 −j wt−n3 , a(ι)t−j = a(ι)ιn4 −j t−n4 . (3.54) (1) , . . . , V (4) ) = a(ι)Xt−1 with Now set Vt = (Vt (1)

Vt

(2)

Vt

(3)

Vt

(4)

Vt

= (a(ι)yt−1 ) , . . . , (a(ι)yt−n1 ) ,

= (a(ι)ut−1 ) , . . . , (a(ι)ut−n2 −1 ) ,

= (a(ι)wt−1 ) , . . . , (a(ι)wt−n3 ) ,

= (a(ι)t−1 ) , . . . , (a(ι)t−n4 )

and

Ut = u , Wt = wt−1+dn , . . . , wt−n , t−1+dn1 , . . . , ut−n2 +1 3 1

Et = t−1+dn1 , . . . , . t−n4 Then (3.54) has a block-matrix representation as ⎛ (1) ⎞ ⎛ Vt ⎞ ⎛ D1 D2 U ⎜ (2) ⎟ t ⎜ D4 0 ⎜ Vt ⎟ ⎜ (3) ⎟ = D ⎝ Wt ⎠ where D = ⎜ ⎝ 0 D5 ⎝ Vt ⎠ E t (4) 0 0 V t

⎞ D3 0 ⎟ ⎟ 0 ⎠ D6

(3.55)

3.5 Mathematical Appendix

51

The block matrix D in (3.55) is a d(n1 + 1 + n2 + n3 + n4 ) × d(3dn1 + 1 + n2 + n3 + n4 ) matrix with constant coeﬃcients. If the rank of D were less than d(n1 + 1 + n2 + n4 + n3 ), then there exist vectors α1 , . . . , αn1 , β1 , . . . , βn2 +1 , γ1 , . . . , γn3 , and δ1 , . . . , δn4 in Rd such that n1

i−1 α = −a(s) i A(s)B(s)s

i=1 n1

i−1 α = −a(s) i A(s)C(s)s

n 2 +1 j=1 n3

γj Id sj−1 ,

i=1

j=1

n1

n4

i−1 α = −a(s) i A(s)D(s)s

i=1

βj Id sj−1 ,

δj Id sj−1 ,

j=1

which is impossible by Lemma 3.1. Hence the matrix in (3.55) has full rank. Noting that zt−1 = (Ut , Wt , Et ) , we can apply Lemma 3.2 to see that there exists ρ > 0 such that T −dn T −dn 1 1 λmin Vt Vt ≥ ρλmin zt zt . t=1

t=1

To complete the proof, it suﬃces to show that ⎛ ⎞ T −dn T dn1 1 λmin Vt Vt ≤ (dn1 + 1) ⎝ a2j ⎠ λmin Xt Xt , t=1

(3.56)

t=1

j=0

Let ζ be a unit vector. Then ⎛ ζ Vt Vt ζ = ⎝

dn1

⎞2

⎛

aj ζ Xt+dn1 −j ⎠ ≤ ⎝

j=0

implies T −dn1 t=1

dn1

⎞ a2j ⎠

j=0

dn1

(ζ Xt+dn1 −j )2

j=0

⎞ ⎛ dn1 T a2j ⎠ ζ Vt Vt ζ ≤ (dn1 + 1) ⎝ ζ Xt Xt ζ, j=0

t=1

proving (3.56).

Proposition 3.5. Let the following hypotheses be satisﬁed. (i) The matrix polynomials A(s), B(s), C(s), and D(s) deﬁned in (3.46) are left coprime and (3.52) holds for all t. (ii) Assumptions 3.1 and 3.2 are fulﬁlled.

52

3 Adaptive Learning in Linear Models

e (iii) The inputs ut = yt,t+1+n satisfy the following conditions. Let Gt := 2 σ(Ft−1 , yt , wt ) be deﬁned as before. Set u˜t = ut − E[ut |Gt ] with Ft measurable ut and assume, in addition, that P-a.s.:

ut α |Gt ] < ∞ for some α > 2, sup E[ ˜ t≥1

λmin

$! T

˜t u ˜ t t=1 u

%

log T + ˜ uT 2 !T t=1

(3.57)

→ ∞ as T → ∞,

$ %% $! T ˜t u ˜

E[ut |Gt ] 2 = O λmin . t t=1 u

(3.58) (3.59)

For arbitrary n > 0, set Zt := ( t+n ; ut+n−1 , wt+n−1 , t+n−1 ; . . . ; ut , wt , t ) .

% Zt Zt % >0 $! lim inf T T →∞ λmin u ˜ u ˜ t t t=1

Then

λmin

$!

T t=1

(3.60)

P − a.s.

The proof of Proposition 3.5 will be prefaced by Theorem 3.4 and Corollary 3.4 which we state without proof. Theorem & Wei 1986b, Thm. 3, p.235) 3.4. (Lai ξt + ηt be a random vector and {Ht }t∈N be an increasing sequence Let ζt = τt of σ-algebras. Suppose that the following properties hold with probability 1. (i) {ηt }t∈N is an Rd2 -valued martingale diﬀerence sequence with respect to {Ht }t∈N such that (a) supt∈N E[ ηt α |H%t ] < ∞ for some α > 2, $! T (b) λmin → ∞ as T → ∞. t=1 ηt ηt (ii) ξt ∈ Rd2 is Ht−1 measurable with T T T 2 .

ξt = O λmin ηt ηt + λmin τt τt t=1

t=1

t=1

(iii) τt ∈ Rd1 is Ht−1 measurable with %% $! $ !T T 2 , t=1 τt = o λmin t=1 ηt ηt $! % T (b) λmin → ∞ as T → ∞. t=1 τt τt

(a) log

3.5 Mathematical Appendix

53

Then $! % T λmin ζ ζ t=1 t t " $! $! %# > 0 % lim inf T T T →∞ , λ min λmin min t=1 τt τt t=1 ηt ηt

P − a.s.

Corollary 3.4. Under the hypotheses of Theorem 3.4, if, in addition, $! % T λmin t=1 τt τt $! % > 0 P − a.s., lim inf (3.61) T T →∞ λmin t=1 ηt ηt then lim inf T →∞

λmin λmin

$! T $!

t=1 ζt ζt

T t=1

ηt ηt

% % >0

P − a.s.

Proof of Proposition 3.5. The proof consists of an induction argument on n and of a repeated application of Theorem 3.4. To apply Theorem 3.4, let n = 1 and proceed in three steps. (1) Step 1. Consider the random vector ζt = (wt , t ) . Set Ht−1 = σ(Ft−1 , t ), τt = t , ξt = E[wt |Ht−1 ], and ηt = wt − E[wt |Ht−1 ]. By Assumptions 3.1 and 3.2, of Corollary 3.4 are fulﬁlled. In particular, since $! all requirements % T = O(T ), u ˜ u ˜ λmin t t t=1

lim inf T →∞

λmin

$! T

(1) (1) t=1 ζt ζt

λmin

$!

T t=1

u˜t u ˜ t

%

%

P − a.s.

>0

(3.62)

Step 2. Consider a second random vector ζt = (ζt , u ) . Set τt = ζt , !T t ξt = E[ut |Gt ], ηt = u ˜t , and Ht = Gt−1 . By Step 1, log t=0 τt 2 = O(log T ). (2) In view of (3.57)-(3.59), ζt again $! meets %all requirements of Corollary 3.4. T = O(T ) that Hence, we conclude from λmin ˜t u ˜ t t=1 u (2)

lim inf T →∞

λmin

$! T

λmin

(2) (2) t=1 ζt ζt

$!

T t=1

u˜t u ˜ t

%

(1)

(1)

% >0

P − a.s.

(3.63)

Step 3. Consider a third random vector ζt = (ζt , ) , where τt = ζt , !Tt+1 ξt = 0, ηt = t+1 , and Ht = Ft+1 . Since again log t=0 τt 2 = O(log T ), it (3) follows from Assumption 3.1 (i) and (ii) that ζt satisﬁes all hypotheses of Corollary 3.4. From an argument analogous to Step 2, we obtain (3)

(2)

(2)

54

3 Adaptive Learning in Linear Models

lim inf T →∞

λmin

$! T

(3) (3)

t=1 ζt ζt $! % T λmin u ˜ u ˜ t=1 t t

% >0

P − a.s.

(3.64)

This proves the case n = 1. By (3.58), we have T T −m λmin u ˜t u ˜t ∼ λmin u ˜t u˜t t=1

t=1

for any ﬁxed m and a repeated application of the above three steps establishes the desired conclusion for arbitrary n.

The key ingredient of the proof of Theorem 3.2 is the following Theorem 3.5 which is an extension of Lai & Wei (1986b, Thm. 5 (ii), p. 242) to a system with additional exogenous variables as inputs. Theorem 3.5. Let the hypotheses of Proposition 3.5 be satisﬁed. Then the following properties hold true P-a.s.: !T !T (a) t=1 ut 2 = O(T ), (b) t=1 ˜ ut 2 = O(T ), % $! T λmin t=1 Xt Xt $! % > 0. (c) lim inf T T →∞ λmin ˜t u ˜ t t=1 u Proof. Property (b) $ follows from%(3.57) as shown in Lai & Wei (1983). This, !T in turn, implies λmin ˜t u ˜ = O(T ) and (a) follows from (3.59). t t=1 u The last property (c) follows from Proposition 3.4 by ﬁrst showing that the regressors (3.36) can be reduced to regressors (3.53) consisting only % of $! T inputs and then applying Proposition 3.5, noting that λmin ≥ t=1 zt zt $! % T λmin for any sub-vector zT of Zt , where the latter was deﬁned t=1 Zt Zt in (3.60). Note in this connection that interchanging any vector entries of a regressor zt will not change the eigenvalues of the matrix zt zt , since it amounts to interchanging columns and rows simultaneously.

Theorem 3.6 is an extension of Lai & Wei (1986b, Thm. 6 (ii)) and provides conditions under which strong consistency of the AML algorithm obtains. Theorem 3.6. Let the following hypotheses be satisﬁed. (i) Assumptions (i)-(iii) of Theorem 3.2 are satisﬁed. (ii) Assumption (iii) of Proposition 3.5 is satisﬁed. Then the AML-algorithm (3.41) generates strongly consistent estimates, i.e., θˆt → θ

P − a.s. as t → ∞.

3.5 Mathematical Appendix

55

! Proof. By Theorem 3.5, Tt=1 ut 2 = O(T ). In view of Assumption (i), all prerequisites of Proposition 3.1 are then satisﬁed, implying that there exists α > 0 such that Xt 2 = O(tα ) P-a.s. Hence T log λmax Xt Xt = O(log T ) P − a.s. t=1

It follows from (3.58) and Theorem 3.5 that $! % T λmin t=1 Xt Xt → ∞ P − a.s. as T → ∞ log T and hence (3.49) is satisﬁed. The assertion then follows from Proposition 3.3.

We are now ready to establish Theorem 3.2. 1

Proof of Theorem 3.2. By Assumption (iii) and (iv) we have wt = o(t 2 ), 1 1 e

= o(t 2 ) P-a.s. Hence by Assumption (i), the system

t = o(t 2 ), and yt,t+j (3.45) is stable in the sense of Proposition 3.1. In order to apply Theorem 3.6 it remains to show that Conditions (3.57)e (3.59) of Proposition 3.5 are fulﬁlled. As before, set ut = yt,t+1+n . Since by 2 (3.44) u ˜t = ut − E[ut |Gt ] = vt , Condition (3.57) is satisﬁed. By an analogous argument to that of Remark 3.7 (ii), Assumption (iv) on the dither vt implies that P-a.s. !T 1 (a) vT = o(T 2 ), (b) t=1 vt 2 = O(T ), $! % T (c) lim inf T1 λmin > 0. t=1 vt vt T →∞

Properties (a) and (b) imply Condition (3.58). Since E[ut |Gt ] = ut − vt ≤ !T ct , it follows from (3.44) that t=1 E[ut |Gt 2 = O(T ). Using Property (c), this yields Condition (3.59). Thus Assumption (ii) of Theorem 3.6 is satisﬁed

and we can apply Theorem 3.6 to conclude that θˆt → θ P-a.s. Proof of Theorem 3.3. Since all eigenvalues A must lie strictly within the unit circle, Assumptions (iii) and (iv) of Theorem 3.2 imply that the system 1 (3.45) is stable in the sense of Cororollary 3.1, i.e., yt = o(t 2 ) P-a.s. In view of Theorem 3.2 it remains to show that all orbits of (3.45) eventually become orbits with rational expectations. It follows from the recursive 1 deﬁnition of the prediction errors (3.40) that ˆ t = o(t 2 ) and t − t = − ˆ

n4 i=1

ˆ t−1 , D(i) (ˆ t−i − t ) + (θ − θˆt )X

t ∈ N.

(3.65)

56

3 Adaptive Learning in Linear Models

By Assumption 3.1 (ii), all eigenvalues of the matrix ⎛ ⎞ −D(1) · · · · · · −D(n4 ) ⎜ Id 0 ··· 0 ⎟ ⎜ ⎟ D=⎜ . ⎟ .. . . .. .. ⎝ .. ⎠ . 0

· · · Id

0

associated with (3.65) lie within the unit circle. Since the additive term in (3.65) vanishes by Assumption (iii), it follows that P-a.s. lim (ˆ t − t ) = 0.

t→∞

As a consequence, there exists ttol > 0 such that ˆ xt , Y e ; θˆt ) = o(t 21 ) P-a.s. for all t ≥ ttol .

ψ(ˆ t−1 Hence, if ctol is chosen large enough, censoring ct will not be applied anymore after time ttol . An an application of Proposition 3.2 to the system under unbiased no-updating (3.20) and its approximation as given by (3.45) implies that all orbits of (3.45) eventually become orbits with rational expectations.

4 Economic Models Subject to Stationary Noise

In this chapter the concepts of Chapter 2 are generalized to the case in which the exogenous perturbations are driven by a stationary ergodic process. We establish necessary conditions for the existence and uniqueness of unbiased forecasting rules. For this purpose it is convenient to imbed our setup into the theory of random dynamical systems. In joining the dynamical systems theory with the theory of stochastic processes, the theory of random dynamical systems developed by Arnold (1998) and others provides a natural formal framework to analyze the behavior of economic systems subject to stationary ergodic perturbations. To apply the theory of random dynamical systems, we extend the sequence of random variables {ξt }t∈N with values in Ξ ⊂ Rdξ deﬁning the random environment of the economy to a sequence {ξt }t∈Z with so-called two-sided time Z := {0, ±1, ±2, . . .}. We will represent the stochastic perturbations by means of a metric dynamical system which essentially amounts to using the equivalent canonical realization process of a stochastic process. To ﬁx the notation, let ϑ : Ω → Ω be a measurable invertible map on a probability space (Ω, F , P) which is measure preserving with respect to P and whose inverse ϑ−1 is again measurable. Assume that P is ergodic with respect to ϑ and let ϑt denote the t-th iterate of the map ϑ. The collection (Ω, F , P, {ϑt }t∈Z ) is called an ergodic metric dynamical system, cf. Arnold (1998, Appendix A.2, p. 542). It is well known that any stationary ergodic process {ξt }t∈Z can be represented by an ergodic metric dynamical system. This means that there exists a measurable map ξ : Ω → Ξ such that for each ﬁxed ω ∈ Ω, a sample path of the noise process is given by ξt (ω) = ξ(ϑt ω),

t ∈ Z.

(4.1)

The following assumption is a reﬁnement of Assumption 2.1 of Chapter 2. Assumption 4.1. Let (Ω, F , P, {ϑt }t∈Z ) be an ergodic metric dynamical system on a complete probability space (Ω, F , P). {ξt }t∈Z is a stationary ergodic

58

4 Economic Models Subject to Stationary Noise

stochastic process on (Ω, F , P) with values in a subset Ξ ⊂ Rdξ which admits a representation of the form (4.1). In this setup, the set Ω may best be thought of the space of all sample paths of the noise process, so that ω ∈ Ω denotes a particular sample path. Let us illustrate this point with a simple and standard example. Example 4.1. Consider the case in which {ξt }t∈Z is a sequence of Ξ-valued iid random variables whose probability distribution is µ. Then (Ω, F , P) = (Ξ Z , B(Ξ)Z , µZ ), where Ω is the space of all sample paths of the process, B(Ξ)Z is the Borel σ-algebra of all cylinder sets, and µZ is the product measure. Each element ω = {ω(s)}s∈Z ∈ Ω is a doubly inﬁnite sequence describing a sample path of the process. The map ϑ : Ω → Ω,

ω → ϑω

is deﬁned by (ϑω)(s) := ω(s + 1), s ∈ Z and is called the left shift such that (ϑt ω)(s) = ω(s + t) for all t, s ∈ Z. If ξ : Ω → Ξ,

ξ(ω) := ω(0)

denotes the evaluation map, a representation of the form (4.1) is obtained. More details are found in Appendix A.2 of Arnold (1998). As before, let X ⊂ Rdx be the space of endogenous variables and xt ∈ X denote the vector of endogenous variables describing the state of the economy at date t. Each xt ∈ X is subdivided into xt = (xt , yt ) ∈ X × Y = X, where yt ∈ Y ⊂ Rdy contains the endogenous variables for which expectations are ¯ formed, xt ∈ X ⊂ Rdx is the vector of the remaining variables. As before, an economic law is a map G : Ξ × X × Ym −→ X

(4.2)

with the interpretation that the state of the economy at time t + 1 is given by e e xt+1 = G(ξ(ϑt ω), xt , yt,t+1 , . . . , yt,t+m ),

where xt is the current observable state of the economy and e e yte := (yt,t+1 , . . . , yt,t+m ) ∈ Ym

is the vector of forecasts for (yt+1 , . . . , yt+m ) formed at time t. Again, xt itself may be a vector of lagged endogenous variables including past realizations ξs , s ≤ t of the exogenous noise process and past forecasts. Remark 4.1. Formally, each map G(ξ(·), x, y e ) : Ω → X,

(x, y e ) ∈ X × Ym

4.1 Forecasting Rules for Stationary Noise

59

is a random variable which is allowed to depend on the whole path ω ∈ Ω. For many economic applications, two cases in which ξt is assumed to be observable at date t are of particular interest. This can be formalized as follows. Setting ξ(ω) = ω(0), ω ∈ Ω as in Example 4.1 with iid noise, one has ξt (ω) = ξ(ϑt ω) in the ﬁrst case such that xt+1 becomes a deterministic function of the observable variables ξt (ω), xt , and yte at date t. Setting ξ(ω) = ω(1), ω ∈ Ω in the second case, one has ξt+1 (ω) = ξ(ϑt ω) and xt+1 (·) = G(ξt+1 (·), xt , yte ) becomes a random quantity in the sense that ξt+1 (ω) is not observable prior to period t + 1.

4.1 Forecasting Rules for Stationary Noise As in the Markovian case, we assume that a forecasting agency which is boundedly rational in the sense of Sargent (1993) is responsible for issuing all relevant forecasts. At date t, the time-series information of the forecasting agency e }s≤t , where includes the actual history of the economy {xs , ys−1 e e e = (ys−1,s , . . . , ys−1,s−1+m ) ∈ Ym , ys−1

s∈Z

(4.3)

e is of particas before. It will turn out that the vector of previous forecasts yt−1 e e ular importance. The forecasts yt,t+1 , . . . , yt,t+m are determined according to a forecasting rule depending on available information. In view of the present setup with stationary ergodic noise, a forecasting rule ψ = (ψ (1) , . . . , ψ (m) ) is formally deﬁned as a function

ψ : Ω × X × Ym −→ Ym ,

(ω, x, y e ) −→ ψ(ω, x, y e ),

(4.4)

such that e e = ψ (i) (ϑt ω, xt , yt−1 ), yt,t+i

i = 1, . . . , m

is the forecast for the future realization yt+i , i = 1, . . . , m, based on information available in period t. To reﬂect the fact that the forecasting rule depends on the history of the economy, let F 0 be a suitably deﬁned sub-σ-algebra of F which describes the past of the economy. This σ-algebra will be speciﬁed in detail in Section 4.1.2 below. Let each forecasting rule ψ(·, x, y e ),

x ∈ X, y e ∈ Ym

be F 0 /B(Ym ) measurable with B(Ym ) denoting the σ-algebra of all Borel sets of Ym . Loosely speaking, this measurability assumption imposed on (4.4) e together with the deﬁnition of ϑ implies that each yt,t+i will depend only on entries up to a certain time index of the path ω = (. . . , ω−1 , ω0 , ω1 , . . .) ∈ Ω. Inserting the forecasting rule (4.4) into the economic law (4.2) yields a discrete-time random dynamical system given by the time-one map

60

4 Economic Models Subject to Stationary Noise

Gψ : Ω × X × Ym −→ X × Ym , deﬁned by

Gψ (ω, x, y e ) =

G(ξ(ω), x, y e )

ψ ϑω, G(ξ(ω), x, y e ), y e

(4.5) .

e Let xt−1 describe the state of the economy at time t − 1 and yt−1 the vector of forecasts based on information available at time t − 1. Then an application of the map (4.5) yields xt e ) = Gψ (ϑt−1 ω, xt−1 , yt−1 yte e ) G(ξ(ϑt−1 ω), xt−1 , yt−1 , = e e ψ(ϑt ω, G(ξ(ϑt−1 ω), xt−1 , yt−1 ), yt−1 )

such that xt describes the state of the economy at time t and yte the vector of forecasts based on information available at that time. The map Gψ governs the evolution of the economy as follows. If the system started in some state (x0 , y0e ), then the iteration of the map (4.5) under the perturbation ω induces a measurable map, again denoted by Gψ , of the form Gψ : N × Ω × X × Ym → X, which is recursively deﬁned by ⎧

⎨Gψ ϑt−1 ω, Gψ (t − 1, ω, x0 , y0e ) if t > 0 e , (4.6) Gψ (t, ω, x0 , y0 ) = ⎩(x , y e ) if t = 0 0 0 such that (xt , yte ) = Gψ (t, ω, x0 , y0e ) is the state of the system at time t. By abuse1 of notation, we write Gψ (1, ω, x0 , y0e ) ≡ Gψ (ω, x0 , y0e ). The map (4.6) generated by the time-one map Gψ describes the random dynamical system over a metric dynamical system (Ω, F , P, {ϑt }t∈Z ) in the sense of Arnold (1998). For arbitrary initial conditions (x0 , y0e ) ∈ X × Ym and any perturbation ω ∈ Ω, the sequence of points γ(x0 , y0e ) := {(xt , yte )}t∈N with (xt , yte ) = Gψ (t, ω, x0 , y0e ), t ∈ N is called an orbit of the random dynamical system Gψ starting at (x0 , y0e ). In the sequel, (4.5) or, equivalently, (4.6) will be referred to as an economic random dynamical system (ERDS). It is easily seen from (4.6) that the map Gψ has two crucial properties. The ﬁrst one is the fact that Gψ (0, ω, x, y e ) = (x, y e )

for all (x, y e ) ∈ X × Ym , ω ∈ Ω

is the identity map on X × Ym . The second property is

Gψ (t2 + t1 , ω, x, y e ) = Gψ t2 , ϑt1 ω, Gψ (t1 , ω, x, y e )

(4.7)

(4.8)

for all ω ∈ Ω, (x, y e ) ∈ X × Ym , and arbitrary t1 , t2 ∈ N. To simplify the exposition and to deﬁne a ﬁltration for the stochastic process (4.6) in Section 1

This is done to avoid an inﬂation of symbols. No confusion should arise.

4.1 Forecasting Rules for Stationary Noise

61

4.1.2, we will assume that for each ω ∈ Ω, the map Gψ (ω, ·) deﬁned in (4.5) is invertible. Writing m m G−1 ψ (ω, ·) : X × Y → X × Y

for the inverse of Gψ (ω, ·), the map (4.6) can then be extended to a measurable map Gψ : Z × Ω × X × Ym → X × Ym , by recursively setting for each negative time t < 0,

t e Gψ (t, ω, x0 , y0e ) = G−1 ψ ϑ ω, Gψ (t + 1, ω, x0 , y0 ) . Then property (4.8) holds for all t1 , t2 ∈ Z and Gψ is said to satisfy the socalled cocycle property. It is shown in Arnold (1998, Thm. 1.1.6, p. 7) that the invertibility of the time-one map (4.5) follows from requiring the cocycle property. We are now ready to characterize those forecasting rules which are correct in the sense that they provide best least-squares predictions conditional on available information and thus rational expectations. 4.1.1 Consistent Forecasting Rules In this section we will develop the notion of a consistent forecasting rule for the case with stationary ergodic noise. Having deﬁned a forecasting rule of the functional form (4.4), assume that the information available in a particular period t ∈ Z is given by the sub-σ-algebra

(4.9) Ft,r := σ Gψ (s, ·, x, y e ) t − r − 1 < s ≤ t, x ∈ X, y e ∈ Ym of F which is generated by the past r + 1 generalizations of the process (4.6). Writing Ft ≡ Ft,∞ for r = ∞, the sequence of σ-algebras {Ft }t∈Z deﬁnes a ﬁltration to which the stochastic process (4.6) is adapted. We will stipulate the length of the memory to a ﬁxed number r ∈ N which may be arbitrary large and which describes the amount of available information. This assumption may be justiﬁed both economically and mathematically. From a mathematical viewpoint, the system may be Markovian as before such that only a ﬁnite past inﬂuences the future states of the system, as discussed in Chapter 2.2 From an economic point of view, there are many reasons why agents use only a ﬁnite memory when taking decisions, one being that their capacity to process information may be limited. In Lemma 4.1 on p. 80 of Section 4.5 we show that Ft,r = ϑ−t F0,r for each t ∈ Z and arbitrary r ∈ N. It is therefore natural to assume from now on that each map ψ(·, x, y e ), x ∈ X, y e ∈ Ym is F0,r /B(Ym ) measurable. Then each ψ (i) (ϑt ·, x, y e ) is Ft,r /B(Ym) measurable. As before, let 2

Recall that if {ξt }t∈Z is iid, then (4.6) is a Markov process, see also Arnold (1998, Sec. 2.1.3, p. 53).

62

4 Economic Models Subject to Stationary Noise e e yt,t+i = ψ (i) (ϑt ω, xt , yt−1 ),

i = 1, . . . , m

(4.10)

denote period-t’s forecasts for the future realizations yt+i , i = 1, . . . , m, respectively. The factorization lemma (Bauer 1992, p. 71) then guarantees the existence of a forecasting rule Ψ = (Ψ (1) , . . . , Ψ (m) ) of the form (2.13) as deﬁned in Chapter 2, given by a map Ψ : X × Σ → Ym , such that for each ω ∈ Ω and each i = 1, . . . , m,

e e e e (4.11) = ψ (i) (ϑt ω, xt , yt−1 ) = Ψ (i) xt , yt−1 , xt−1 , . . . , yt−r , xt−r yt,t+i P-a.s., where

(xs , yse ) = Gψ (s, ω, x0 , y0e ),

s ≤ t − 1.

In view of the estimation techniques which will be introduced in Chapter 5, this observation shows that a forecasting rule of the functional form (4.10) can always be written as a function depending on the history of observable quantities. The more formally deﬁned forecasting rule (4.10) itself is a stationary object, whereas the form (4.11) can be thought as actually being applied by a forecasting agency within the model. Such a forecasting agency, naturally, would not be assumed to know the whole path ω ∈ Ω. Recall that in each period t there are m − 1 forecasts e e , . . . , yt−1,t−1+m yt−1,t+1 e for yt+1 which were set prior to time t, where yt−i,t+1 is based on information available in period t − i and hence Ft−i measurable. As before, let E[·|Ft−i ] denote the conditional expectations operator with respect to Ft−i . All forecast e , i = 0, . . . , m − 1 vanish on average conditional on the errors yt+1 − yt−i,t+1 respective σ-algebras Ft−i if and only if

e ) Ft−i = 0 P − a.s. (4.12) E (yt+1 − yt−i,t+1

for all i = 0, . . . , m − 1. The law of iterated expectations (2.19) implies that e for yt+i , which is chosen in period t, has to satisfy the the forecast yt,t+i consistency condition e e E[yt,t+i |Ft−1 ] = yt−1,t+i

P − a.s.

(4.13)

Since the consistency condition (4.13) has to hold for all i = 1, . . . , m − 1 in all periods t, the law of iterated expectations implies e e |Ft−i ] = yt−i,t+1 E[yt,t+1

P − a.s.

(4.14)

for all i = 1, . . . , m − 1 and all t ∈ N. Hence, if the consistency condition (4.13) holds for all forecasts in all periods prior to time t, then (4.12) is satise ﬁed, provided that the most recent forecast yt,t+1 has a vanishing conditional forecast error, that is, if

4.1 Forecasting Rules for Stationary Noise

63

e E (yt+1 − yt,t+1 ) Ft = 0 P − a.s. In terms of forecasting rules ψ = (ψ (1) , . . . , ψ (m) ) of the form (4.4), the consistency conditions (4.13) can be expressed as follows. Let Ej [·] ≡ E[·|Fj,r ] denote the conditional expectations operator with respect to the σ-algebra Fj,r , where the length of the memory r may be arbitrarily long. In terms of a forecasting rule ψ, the consistency condition (4.13) is now rephrased as e Et−1 [yt,t+i ](ω) = Et−1 ψ (i) (ϑt ·, Gψ (t − 1, ·, x0 , y0e ) (ω) e = yt−1,t+i

P − a.s.

(4.15)

e for each i = 1, . . . , m − 1, t ∈ N, where (xt−1 , yt−1 ) = Gψ (t − 1, ω, x0 , y0e ). Again, the requirement (4.15) states that for each i = 1, . . . , m − 1, the exe pected value of the new forecast yt,t+i conditional on Ft,r has to coincide with e the respective old forecast yt−1,t+i . Since Ft,r ⊂ Ft , condition (4.15) implies (4.13) and thus is weaker than the original consistency condition (4.13). We seek a consistency notion which is time invariant. Recall to this end that by Lemma 4.1 of Section 4.5 Ft,r = ϑ−t F0,r . Proposition 4.1 on page 81 in Section 4.5 then implies that e e e Et−1 [yt,t+i ](ω) = Et−1 ψ (i) (ϑt ·, G(ξ(ϑt−1 ·), xt−1 , yt−1 ), yt−1 ) (ω) e e ), yt−1 ) (ϑt−1 ω), (4.16) = E0 ψ (i) (ϑ·, G(ξ(·), xt−1 , yt−1 e ) = Gψ (t − 1, ω, x0 , y0e ). Combining (4.15) with (4.16), the where (xt−1 , yt−1 desired notion of a consistent forecasting rule now takes the following form.

Deﬁnition 4.1. A forecasting rule ψ = (ψ (1) , . . . , ψ (m) ) of type (4.4) is called (locally) consistent with respect to the sub-σ-algebra F0,r of F if there exists a random set 3 U = {U(ω)}ω∈Ω , U(ω) ⊂ X × Ym , such that for each i = 1, . . . , m − 1, E ψ (i) (ϑ·, G(ξ(·), x, z), z) F0,r (ω) = z (i+1) P − a.s. for all (x, z) ∈ U(ω), where z = (z (1) , . . . , z (m) ). ψ is called globally consistent if U(ω) = X × Ym P-a.s. A consistent forecasting rule as deﬁned in Deﬁnition 4.1 is designed to satisfy (4.13) by construction on a possibly restricted random subset of the state space X×Ym . The deﬁnition contains a minimum requirement for a forecasting rule of the form (4.4) in order to generate best least-squares predictions. As a consequence of the law of iterated expectations, a forecasting rule that is not consistent cannot provide best least-squares predictions. Notice that the consistency requirement includes the case with inﬁnite memory F0,∞ . 3

A random set is simply a set-valued random variable U : Ω → P(X × Ym ), where P(X × Ym ) denotes the space of all subsets of X × Ym , see Arnold (1998, Sec. 1.6, p. 32) for more details.

64

4 Economic Models Subject to Stationary Noise

Example 4.2. As already seen in Example 2.1, the simplest example of a consistent forecasting rule is a forecasting rule that never updates previous forecasts. A no-updating forecasting rule ψ = (ψ (1) , . . . , ψ (m) ) is formally deﬁned by setting for each i = 1, . . . , m − 1, ψ (i) (ω, x, z) := z (i+1)

for all (x, z) ∈ X × Ym , ω ∈ Ω

(4.17)

and letting ψ (m) be an arbitrary function depending on the history of the e process. If xt ∈ X is the current state of the economy at date t and yt−1 = e e , . . . , yt−1,t−1+m ) the vector of forecasts formed in the previous period (yt−1,t e e t − 1, then the current forecast yte = (yt,t+1 , . . . , yt,t+m ) is e e = yt−1,t+i , i = 1, . . . , m − 1, yt,t+i e e yt,t+m = ψ (m) (ϑt ω, xt , yt−1 ).

The forecasting rule (4.17) thus never updates forecasts formed at previous dates. 4.1.2 Unbiased Forecasting Rules To see which forecasting rules generate best least-squares predictions, recall that

yt+1 = g ξ(ϑt ω), xt , yte . Invoking Proposition 4.1 (see Section 4.5, p. 81 below), the expected value of the future state yt+1 reads

Et [yt+1 ](ω) = Et g(ξ(ϑt ·), xt , yte ) (ω)

(4.18) = E0 g(ξ(·), xt , yte ) (ϑt ω) P − a.s., where (xt , yte ) = Gψ (t, ω, x0 , y0e ). Given a vector of forecasts yte ∈ Ym , the forecast error for yt+1 reads e e = g(ξ(ϑt ω), xt , yte ) − yt,t+1 , yt+1 − yt,t+1

while the forecast error conditional on information available at date t is

e e Et (yt+1 − yt,t+1 ) (ω) = E0 g(ξ(·), xt , yte ) (ϑt ω) − yt,t+1 . (4.19) Using (4.18), this gives rise to the deﬁnition of a (conditional) mean error function associated with the economic law G. Deﬁnition 4.2. Let G = (G, g) be an economic law. The (conditional) mean error function of the economic law G with respect to a given sub-σ-algebra F0,r is a function eEG : Ω × X × Ym → Rdy , deﬁned by

eEG (ω, x, z) = E g ξ(·), x, z F0,r (ω) − z (1) , (4.20) where z = (z (1) , . . . , z (m) ).

4.1 Forecasting Rules for Stationary Noise

65

Given an arbitrary state of the economy xt ∈ X at some date t, the mean error function describes all possible mean forecast errors conditional on all information available at date t, regardless of which forecasting or learning e e , . . . , yt,t+m ), the rule has been applied. Given arbitrary forecasts yte = (yt,t+1 conditional forecast error (4.19) vanishes if and only if e ](ω) = eEG (ϑt ω, xt , yte ) = 0 Et [yt+1 − yt,t+1

P − a.s.

(4.21)

Geometrically, condition (4.21) is characterized by the zero-contour set WG = {WG (ω)}ω∈Ω of the error function (4.20), given by the family of subsets " # WG (ω) := (x, y e ) ∈ X × Ym eEG (ω, x, y e ) = 0 , ω ∈ Ω. The zero-contour set WG , also referred to as constraint variety of the economic law G, describes all states of the economy (ω, x, y e ) for which the conditional forecast error (4.19) vanishes. Let · be some norm on,Rdy and- ε ≥ 0 be a ε ε non-negative number. A random neighborhood WG := WG (ω) ω∈Ω of the E zero-contour set of the error function eG is deﬁned by setting ' # " ' ε (ω) := (x, y e ) ∈ X × Ym 'eEG (ω, x, y e )' ≤ ε , ω ∈ Ω. WG 0 For ε = 0, we have WG ≡ WG . The constraint variety imposes a geometric constraint for an unbiased forecasting rule which will be deﬁned next.

Deﬁnition 4.3. Let G be an economic law. A forecasting rule ψ = (ψ (1) , . . . , ψ (m) ) of type (4.4) is called locally ε-unbiased for G with respect to F0,r if there exists a random set U = {U(ω)}ω∈Ω , U(ω) ⊂ X × Ym such that G(ξ(ω), x, y e ) e ε

∈ WG Gψ (ω, x, y ) = (ϑω) ψ ϑω, G(ξ(ω), x, y e ), y e for all (x, y e ) ∈ U(ω), ω ∈ Ω. For ε = 0, locally ε-unbiased forecasting rules are called locally unbiased. Let ψ = (ψ (1) , . . . , ψ (m) ) be locally ε-unbiased in the sense of Deﬁnition e ) ∈ U(ϑt−1 ω) for some time t − 1. Then 4.3 and (xt−1 , yt−1 xt e ε = Gψ (ϑt−1 ω, xt−1 , yt−1 ) ∈ WG (ϑt ω), yte implying that the conditional forecast error for e e yt,t+1 = ψ (1) (ϑt ω, xt , yt−1 )

is less than ε, i.e., ' ' ' ' e 'Et [yt+1 − yt,t+1 ](ω)' = 'eEG (ϑt ω, xt , yte )' ≤ ε P − a.s.

(4.22)

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4 Economic Models Subject to Stationary Noise

In other words, the forecasting rule ψ gives an approximate solution to (4.21) e in the sense that yt,t+1 , up to an error of size ε, is the best least-squares pre-t , e . If, in addition, a locally diction for yt+1 given the history xs , ys−1 s=t−r+1 unbiased forecasting rule is consistent, then (4.22) implies ' ' '

' e e 'Et−i [yt+1 − yt−i,t+1 ](ω)' = 'Et−i Et [yt+1 − yt,t+1 ] (ω)' ≤ ε P − a.s. e for all i = 1, . . . , m − 1, so that the forecast errors of all forecasts yt−i,t+1 , i = 0, . . . , m − 1 is less that ε. This justiﬁes the terminology locally ε-unbiased forecasting rule.

Let γ(x0 , y0e ) = {xt , yte }t∈N with (xt , yte ) = Gψ (t, ω, x0 , y0e ) denote an orbit of the time-one map Gψ corresponding to the economic law G and the forecasting rule ψ which starts at (x0 , y0e ) ∈ X × Ym . A main problem with locally ε-unbiased forecasting rules is that the forecasting rule may well lose its property of being unbiased along such an orbit. This problem can be ruled out by imposing the following additional requirement. Let ψ be a forecasting rule that is locally ε-unbiased on a random set U in the sense of Deﬁnition 4.3 ε (ω) for all ω ∈ Ω. Then the requirement that U and assume that U(ω) ⊂ WG is forward invariant under Gψ , i.e., Gψ (ω, x, y e ) ∈ U(ϑω)

for all (x, y e ) ∈ U(ω), ω ∈ Ω

(4.23)

is suﬃcient for the existence of unbiased orbits. In this case, any orbit γ(x0 , y0e ) ε starting at (x0 , y0e ) ∈ U(ω) satisﬁes (xt , yte ) ∈ U(ϑt ω) ⊂ WG (ϑt ω) for all times t and for all perturbations ω and for this reason will be an ε-unbiased orbit. In other words, an ε-unbiased orbit is an orbit along which all forecasts are εunbiased. For ε = 0, ε-unbiased orbits are referred to as unbiased orbits. These correspond precisely to rational expectations equilibria (REE) in the classical sense. A forecasting rule that generates ε-unbiased orbits will be referred to as unbiased forecasting rule and is deﬁned as follows. Deﬁnition 4.4. Let G = (G, g) be an economic law. A forecasting rule ψ = (1) (m) (ψ , . . . , ψ ) of type (4.4) is called ε-unbiased for G with respect to F0,r if there exists a random set U = {U(ω)}ω∈Ω with U(ω) ∈ X × Ym , such that the following conditions hold: (i) Consistency. For each i = 1, . . . , m − 1, (i) E ψ (ϑ·, G(ξ(·), x, z), z) F0,r (ω) = z (i+1)

P − a.s.

for all (x, z) ∈ U(ω), where z = (z (1) , . . . , z (m) ). (ii) Local Unbiasedness. There exists a non-negative number ε ≥ 0 such that G(ξ(ω), x, z) ε

Gψ (ω, x, z) = ∈ WG (ϑω) ψ ϑω, G(ξ(ω), x, z), z for all (x, z) ∈ U(ω), ω ∈ Ω.

4.1 Forecasting Rules for Stationary Noise

67

ε (iii) Invariance. U(ω) ⊂ WG (ω) for all ω ∈ Ω and

Gψ (ω, x, z) ∈ U(ϑω)

for all (x, z) ∈ U(ω), ω ∈ Ω.

For ε = 0, ε-unbiased forecasting rules are called unbiased. Deﬁnition 4.4 implies that a forecasting rule ψ is ε-unbiased if and only if all orbits γ(x0 , y0e ) starting in (x0 , y0e ) ∈ U(ω) of Gψ are ε-unbiased orbits. While Conditions (i) and (ii) state that the forecasts are locally unbiased, it is the invariance condition (iii) that guarantees that the forecasts of ψ remain unbiased through the course of time. A forecasting rule that generates ε-unbiased orbits for arbitrary initial values x0 ∈ X is deﬁned as follows. Deﬁnition 4.5. Under the hypotheses of Deﬁnition 4.4, a forecasting rule ψ of type (4.4) is called globally ε-unbiased for G with respect to F0,r if, in addition, the following conditions holds true: (iv) Global existence. For each initial state x0 ∈ X and each perturbation ω ∈ Ω, there exists a vector of forecasts y0e ∈ Ym such that (x0 , y0e ) ∈ U(ω). The conditions in Deﬁnitions 4.4 and 4.5 are clearly quite demanding. In view of economic applications and the learning issue treated in these notes, it will often be suﬃcient to have good forecasts in the long run. We will therefore relax the conditions of Deﬁnition 4.4 when we discuss the long-run behavior of a random dynamical system in Section 4.4 and conclude this section with two remarks. Remark 4.2. A particular case arises when the g-part of the economic law (4.2) is F0,r measurable. Then the mean error function takes the form

eEG (ω, x, z) = g ξ(ω), x, z − z (1) and coincides essentially with the deterministic error function in B¨ ohm & Wenzelburger (1999, 2002). An unbiased no-updating forecasting rule would generate perfect foresight in the classical sense. We will make use of this observation in Chapter 6. Remark 4.3. For a ﬁnite memory r < ∞, the factorization lemma and Proposition 4.1 below yield the existence of an error function EG : Ym × X × Σ → Rmdy as deﬁned in Chapter 2, such that for each t ∈ N and each ω ∈ Ω e , xt−r ) eEG (ϑt ω, xt , yte ) = EG (yte , xt , . . . , yt−r

P − a.s.

along any orbit {xt , yte }t∈N .This implies that an unbiased forecasting rule ψ in the sense of Deﬁnition 4.4 induces an unbiased forecasting rule

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4 Economic Models Subject to Stationary Noise

Ψ : X × Σ → Ym of the form (4.11), such that e e e e EG (Ψ (xt , yt−1 , . . . , yt−r , xt−r ), xt , yt−1 , xt−1 , . . . , yt−r , xt−r ) = 0

on a suitably deﬁned subset of X × Σ.

4.2 Existence of Unbiased Forecasting Rules At ﬁrst sight, the problem of existence and uniqueness of unbiased no-updating forecasting rules for systems with an expectational lead looks much like the deterministic case treated in B¨ ohm & Wenzelburger (2004). This is due to the fact that the mean error function deﬁned in Deﬁnition 4.2 is a static and hence time-invariant object the properties of which decide whether or not unbiased forecasting rules exist. As in the deterministic case, the existence and uniqueness problem is encoded in the geometry of the zero-contour set of the error function. The stochastiﬁcation of the deterministic case, however, requires some subtle adjustments which will be developed in this section. The existence of unbiased forecasting rules is based on the following geometric observation. Let V = {V(ω)}ω∈Ω with V(ω) ⊂ X × Ym−1 , ω ∈ Ω be a random set and assume that for each ω ∈ Ω, there exists a function H(ω, ·) : V(ω) → Y such that

eEG ω, x, η, H(ω, x, η) = 0 for all (x, η) ∈ V(ω) P − a.s. (4.24) Assume, in addition, that each H(·, x, η) is F0,r /B(Y) measurable. The idea now is to construct an unbiased no-updating forecasting rule (1)

(m)

ψ = (ψ , . . . , ψ

) : Ω × X × Ym → Ym ,

(4.25)

(i)

using the function H. Since in this case all ψ , i = 1, . . . , m − 1, are projec(m) tions, we are left to deﬁne the forecasting rule ψ . To this end, let xt ∈ X be the current state of the economy at date t and e e e = (yt−1,t , . . . , yt−1,t−1+m ) the vector of forecasts formed in the previous yt−1 period t − 1 such that

e e xt , (yt−1,t+1 , . . . , yt−1,t−1+m ) ∈ V(ϑt ω). e e The forecast yte = (yt,t+1 , . . . , yt,t+m ) formed in period t can now be obtained by setting (i)

e e e yt,t+i = ψ (ϑt ω, xt , yt−1 ) := yt−1,t+i , i = 1, . . . , m − 1,

(m) e e e yt,t+m = ψ (ϑt ω, xt , yt−1 ) := H ϑt ω, xt , pr−1 yt−1 ) ,

where

4.2 Existence of Unbiased Forecasting Rules

pr−1 : Ym → Ym−1 ,

69

(z (1) , . . . , z (m) ) → (z (2) , . . . , z (m) ).

This deﬁnes ψ (ω, ·) on V(ω) × Y for each perturbation ω ∈ Ω. Outside of V, (m) ψ may be an arbitrary function. Since ψ never updates forecasts formed at previous dates, it is a consistent rule in the sense of Deﬁnition 4.1. Moreover, ψ is locally unbiased in the sense of Deﬁnition 4.3, because the function H was assumed to satisfy (4.24). For this reason the conditional forecast error vanishes, that is,

e e ](ω) = eEG ϑt ω, xt , ψ (ϑt ω, xt , yt−1 ) = 0 P − a.s. Et [yt+1 − yt,t+1 By construction of (4.25) all forecasts for yt+1 fulﬁll e ] = 0 P − a.s., Et [yt+1 − yt−i,t+1

i = 0, . . . , m − 1,

(4.26)

as long as the no-updating rule (4.25) has been applied for the past m periods. This shows that ψ∗ is locally unbiased in the sense of Deﬁnition 4.3, provided that (4.24) holds. The question of existence of an unbiased no-updating rule can now be addressed as follows. Consider the function EG induced by eEG , which is deﬁned by Ω × X × Ym −→ Ω × X × Ym−1 × Rdy

, EG : (4.27) (ω, x, z) −→ ω, x, pr−m z, eEG (ω, x, z) where pr−m : Ym → Ym−1 ,

(z (1) , . . . , z (m) ) → (z (1) , . . . , z (m−1) ).

If the induced error function (4.27) is globally invertible, then (4.27) has an inverse function H which must be of the functional form Ω × X × Ym−1 × Rdy → Ω × X × Ym−1 × Y

H: (4.28) (ω, x, η, ζ) → ω, x, η, H0 (ω, x, η, ζ) with some function H0 : Ω × X × Ym−1 × Rdy → Y. If the inverse function H exists, then

eEG ω, x, η, H0 (ω, x, η, 0) = 0 for all (ω, x, η) ∈ Ω × X × Ym−1 . Deﬁning H(ω, x, η) := H0 (ω, x, η, 0),

(ω, x, η) ∈ Ω × X × Ym−1 ,

this implies

eEG ω, x, η, H(ω, x, η) = 0,

(ω, x, η) ∈ Ω × X × Ym−1 .

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4 Economic Models Subject to Stationary Noise

If V(ω) = X × Ym−1 , ω ∈ Ω, then the forecasting rule ψ as deﬁned in (4.25) is unbiased in the sense of Deﬁnition 4.4 globally on X × Ym . The question whether unbiased no-updating rules of the form (4.25) exist is therefore reduced to the question whether the induced error function (4.27) is invertible. Invoking the Inverse Function Theorem or, equivalently, the Implicit Function Theorem, it is now possible to provide suﬃcient conditions for the existence of a unique unbiased no-updating forecasting rule for an economic law G. Theorem 4.1. Let G = (G, g) be an economic law. Suppose that the following conditions hold: (i) There exists a F0,r /B(X) measurable random variable x0 : Ω → X and a F0,r /B(Ym ) measurable random variable z0 : Ω → Ym , such that

eEG ω, x0 (ω), z0 (ω) = 0 P − a.s. (ii) For each ω ∈ Ω, the function h(ω, x, z) := E[g(ξ(·), x, z)|F0,r ](ω) is continuously diﬀerentiable with respect to x∈ X and z = (z (1) , . . . , z (m) ) ∈ Ym ; the derivative Dz(m) h ω, x0 (ω), z0 (ω) is invertible for all ω ∈ Ω, where Dz(m) denotes the derivative with respect to z (m) . Then there exists a family of open sets V = {V(ω)}ω∈Ω with V(ω) ⊂ X×Ym−1 , ω ∈ Ω and functions H(ω, ·) : V(ω) → Y, ω ∈ Ω satisfying

(m) H ω, x0 (ω), pr−1 z0 (ω) = z0 (ω),

ω ∈ Ω,

such that

eEG (ω, x, η, H ω, x, η) = 0

for all (x, η) ∈ V(ω)

P − a.s.

Moreover, the no-updating rule ψ given in (4.25) is locally unbiased in the sense of Deﬁnition 4.3 and each ψ (·, x, y e ), (x, y e ) ∈ X × Ym is F0,r /B(Ym ) measurable. Proof. The existence of the map H and thus of ψ follows from a ω-wise application of the Implicit Function Theorem, e.g., see Lang (1968, p. 355). The measurability property of ψ follows from the fact that each eEG (·, x, z), is F0,r /B(Rdy ) measurable.

(x, z) ∈ X × Ym

To formulate a global existence theorm we will invoke what in the mathematical literature is known as a Global Inverse Function Theorem. Recall to

4.2 Existence of Unbiased Forecasting Rules

71

this end the notion of a simply connected set. Loosely speaking, a simply connected set contains no holes. More formally, a set X ⊂ Rd is simply connected if every closed path p : [0, 1] → X, p(0) = p(1) is homotopic to a constant path, that is if there exists a continuous function P : [0, 1] × [0, 1] → X with P (t, 0) = p(t) and P (t, 1) ≡ x0 ∈ X for all t ∈ [0, 1], cf. Lipschutz (1965, p. 186). Simple examples for simply connected sets are X = Rdx and Y = Rdy . Theorem 4.2. Under the hypotheses of Theorem 4.1, assume that the following conditions hold: (i) For each ω ∈ Ω, h(ω, ·) ∈ C 1 (X × Ym , Y) and Dz(m) h(ω, x, z) is invertible for all (ω, x, z) ∈ Ω × X × Ym . (ii) For each ω ∈ Ω, there exist positive constants α(ω) and β(ω) such that . EG (ω, x, z) + Dz(m) h(ω, x, z)−1 ≤ α(ω) (x, z) +β(ω)

Dz(m) h(ω, x, z)−1 De . denotes the derivative with respect P-a.s. for all (x, z) ∈ X × Ym , where D (1) (m−1) to the variables (x, z , . . . , z ). (iii) X and Y are simply connected. Then all conclusions of Theorem 4.1 hold and a unique no-updating rule ψ : Ω × X × Ym → Ym as deﬁned in (4.25) exists, which is unbiased in the sense of Deﬁnition 4.5, globally on X × Ym . Proof. We only have to assure that (4.28) exists. The derivative of EG with respect to (x, z) ∈ X × Ym takes the (block matrix) form I 0 . DEG (ω, x, z) = . E DeG (ω, x, z) Dz(m) h(ω, x, z) Indeed, DEG (ω, x, z) is invertible if and only if Dz(m) h(ω, x, z) is invertible. The inverse is I 0 DEG (ω, x, z)−1 = . E (ω, x, z) Dz(m) h(ω, x, z)−1 . −Dz(m) h(ω, x, z)−1 De G Using the respective matrix norms, . E (ω, x, z) + Dz(m) h(ω, x, z)−1 .

DEG (ω, x, z)−1 ≤ 1+ Dz(m) h(ω, x, z)−1 De G Conditions (i) and (ii) are precisely the two hypotheses needed for a Global Inverse Function Theorem (Deimling 1980, Thm. 15.4, p. 153) to be applicable, whereas Condition (iii) is precisely that additional requirement needed for the case in which X and Y are not vector spaces. This guarantees the existence of a unique globally deﬁned inverse function H ≡ E−1

G .

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4 Economic Models Subject to Stationary Noise

It is straightforward to generalize Theorem 4.2 to the case of ε-unbiased forecasting rules. Theorem 4.2 reduces the problem of existence and uniqueness of an unbiased forecasting rule to the problem of global invertibility of the induced error function. Condition (i) in Theorem 4.2 provides a local invertibility criterion for the induced error function (4.27), whereas (ii) is an upper bound suﬃcient for the existence of a unique global inverse and (iii) is a topological criterion. Notice that for the existence of an unbiased forecasting rule it suﬃces to have invertibility of (4.27) on an open neighborhood of X × Ym−1 × {0} in the image of EG . By virtue of the Inverse Function Theorem (see also Gale & Nikaido 1965), locally as well as globally unbiased forecasting rules are uniquely determined by the properties of WG and thus by the fundamentals of the economy. This stipulates the functional form of an unbiased forecasting rule. R dy

R dy

ε+ W(ω) ε−

ε+ W(ω) ε− (a) Uniqueness.

X × Ym

(b) Multiplicity.

X × Ym

Fig. 4.1. Constraint varieties.

If the induced error function (4.27) is not globally invertible, it may happen that there exist multiple local inverses and thus several locally unbiased no-updating forecasting rules. Uniqueness and existence of unbiased forecasting rules are therefore encoded in the geometry of the zero-contour set WG , as graphically stylized in Figure 4.1. Topological methods to analyze the structure of WG along with a global implicit function theorem can, for instance, be found in Rheinboldt (1969). However, even if all conditions of a theorem like Theorem 4.2 hold, for economic laws G = (G, g) with non-linear g–parts, it will, in general, be impossible to construct an unbiased forecasting rule ψ analytically. Recall the remarkable feature of economic systems with expectational leads (m > 1). An unbiased no-updating forecasting rule (4.25), if it exists, generates rational expectations in the sense that all forecast errors (4.26) conditional on information available at date t vanish, including those for forecasts which have been made prior to that date, see also Theorem 2.1 of Chapter 2. e The no-updating rule (4.25) is designed in such a way that Et [yt+1 ] = yt,t+1 e holds essentially for all choices of yt,t+i , i = 1, . . . , m − 1. The idea is to e e choose yt,t+m in such a way that the forecast error for yt,t+1 is minimal. This implies a considerable freedom in selecting the ﬁrst m−1 forecasts because any

4.3 MSV Predictors

73

martingale diﬀerence sequence could, in principle, be added to these forecasts without violating the unbiasedness. Apart from the geometry of WG , this fact constitutes a primary cause for the emergence of multiple unbiased forecasting rules and thus of multiple rational expectations equilibria in the presence of expectational leads.

4.3 MSV Predictors In linear models the notion of minimal-state-variable solutions introduced by McCallum (1983, 1998, 1999) is a popular concept to describe situations in which agents have rational expectations along particular trajectories of the system. In Chapter 3, Section 3.2.2, we showed in a linear context that MSV predictors may be seen as generators for minimal-state-variable solutions. We will now specialize the notion of a MSV predictor to non-linear economic laws of the form (4.2). By virtue of Theorem 4.1, the existence of a set U on which unbiased forecasts are possible depends solely on the economic law G. This holds in particular for any subset V ⊂ U which is forward invariant under Gψ because the restriction ψ |V of a locally unbiased forecasting rule is uniquely determined by WG . Hence, the system Gψ when restricted to the invariant set V is already determined by the economic fundamentals and, in principle, may exhibit any type of dynamic behavior. As pointed out in the previous section, initial conditions have to lie in a forward-invariant set V of Gψ in order to guarantee rational expectations along the whole orbit. e There is a considerable freedom in choosing the initial forecasts y0,i i= 1, . . . , m, for an initial state x0 ∈ X of the economy. In particular, diﬀerent initial forecasts could, in principle, lead to diﬀerent long-run outcomes of the economy even under rational expectations. To eliminate this degree of freedom, we will derive forecasting rules from a random dynamical system as follows. Let φ be a random dynamical system on X over a metric dynamical system (Ω, F , P, {ϑt }t∈Z ), given by xt+1 = φ(1, ϑt ω, xt ) = φ(t + 1, ω, x0 ),

x0 ∈ X.

(4.29)

One may assume here that (4.29) describes a perceived law of motion in which a forecasting agency believes. Let V := {V(ω)}ω∈Ω be a random set which is forward invariant under φ, i.e., φ(1, ω, x) ∈ V(ϑω), and

for all x ∈ V(ω)

Ftφ := σ φ(s, ·, x) s ≤ t, x ∈ X , t ∈ Z (i)

be sub-σ-algebras pertaining to (4.29). Deﬁne forecasting rules ϕX : Ω × X → X, i = 1, . . . , m, by setting

74

4 Economic Models Subject to Stationary Noise (i)

ϕX (ω, x) := E[φ(i, ·, x)|F0φ ](ω),

i = 1, . . . , m.

(4.30)

As before, let xt ∈ X denote the state of the economy at date t. Using (4.30), we let (i) xet,t+i := ϕX (ϑt ω, xt ), i = 1, . . . , m be the period-t forecasts for xt+i , i = 1, . . . , m, respectively, which are derived from the perceived law (4.29). By Proposition 4.1 (see Section 4.5 below), xet,t+i = E[φ(i, ϑt ·, xt )|Ftφ ](ω) = E[xt+i |Ftφ ](ω) P − a.s.

(4.31)

for all i = 1, . . . , m. If xt+i is thought of being generated from the perceived law (4.29), then the forecasts xet,t+i , i = 1, . . . , m, become the best leastsquares predictions, given the history of realizations {xs }s≤t . By (4.31) and the law of iterated expectations, each forecast derived from (4.30) satisﬁes (i+1)

φ φ ](ω) = E[xt+i |Ft−1 ](ω) = ϕX E[xet,t+i |Ft−1

(1)

(ϑt−1 ω, xt−1 )

(4.32)

(m)

for all i = 1, . . . , m. This implies that ϕX = (ϕX , . . . , ϕX ) is a consistent forecasting rule in the sense of Deﬁnition 4.1. To construct a forecasting rule for an economic law G as deﬁned in (4.2), let πYX : X → Y, (x, y) → y denote the natural projection onto Y. For each i = 1, . . . , m, let

(i) e (4.33) = ϕ(i) (ϑt ω, xt ) := πYX ϕX (ϑt ω, xt ) yt,t+i be the forecast for yt+i , i = 1, . . . , m, respectively. Since the forecasting rule ϕ := (ϕ(1) , . . . , ϕ(m) ) deﬁned in (4.33) uses a minimal amount of endogenous variables, it will be referred to as a minimal-state-variable predictor (MSV predictor) for an economic law G of the form (4.2). It follows from (4.32) that ϕ is a consistent forecasting rule in the sense of Deﬁnition 4.1. ϕ is locally unbiased in the sense of Deﬁnition 4.3 if

(4.34) eEG ω, x, ϕ(ω, x) = 0 for all x ∈ V(ω) P − a.s. For each ω ∈ Ω, let Γϕ (ω) := {(x, z) ∈ V(ω) × Ym | z = ϕ(ω, x)} denote the graph of the map ϕ(ω, ·) and denote by Gϕ the time-one map pertaining to the economic law G and the MSV predictor ϕ. The predictor ϕ is then unbiased in the sense of Deﬁnition 4.4 if Γϕ is forward invariant under Gϕ . Notice again that (4.34) incorporates an invertibility condition of the error function as discussed in Section 4.2. A particular case arises when

φ(1, ω, x) = G ξ(ω), x, ϕ(ω, x) , for all x ∈ V(ω) P − a.s. (4.35)

4.4 Random Fixed Points

75

Then the additional degree of freedom in the unbiasedness condition (4.34), which requires coincidence of objective and subjective ﬁrst moments only, is eliminated and each orbit of the perceived law (4.29) starting in V(ω) corresponds to an orbit of Gϕ . These orbits correspond precisely to minimal-statevariable solutions in the sense of McCallum (1983). The three relationships (4.29), (4.33), and (4.35) may also be seen to deﬁne a functional rational expectations equilibrium in the sense of Spear (1988). This observation allows us to interpret minimal state variable solutions and functional rational expectations equilibria as being generated from a particular kind of unbiased forecasting rules, referred to as unbiased MSV predictors. The two conditions (4.33) and (4.35) constitute a nonlinear equation on an appropriately deﬁned function space for a time-one map (4.29) which determines functional rational expectations equilibria (minimal state variable solutions). In general, such an equation will be much harder to solve than ﬁnding the inverse of an error function on a ﬁnite-dimensional space. Recently, B¨ ohm, Kikuchi & Vachadze (2006) used a numerical method to comupte a MSV predictor for a particular model. Unbiased MSV predictors will most likely not be uniquely determined, as the analysis of the linear case treated in Chapter 3 indicates. However, they retain their importance as a possibility of singling out unbiased forecasting rules under which the resulting rational-expectations dynamics are stable.

4.4 Random Fixed Points For many economic applications it suﬃces to have good forecasts in the long run, that is, ε–unbiased forecasts on special (random) sets or along special orbits of the system. This is the case when the set {U(ω)}ω∈Ω appearing in Deﬁnition 4.4 contains a random attractor. Random attractors are the random analogue of attractors for deterministic dynamical systems (see Arnold 1998, p. 483). Each orbit starting from the corresponding domain of attraction will then eventually become an ε-unbiased orbit. Otherwise an orbit may lose this property. Typical candidates for these special orbits are generated by asymptotically stable random ﬁxed points of an ERDS. The following deﬁnition of a random ﬁxed point (Schmalfuß 1996, 1998)4 includes a stability notion given as Deﬁnition 7.4.6 in Arnold (1998). Deﬁnition 4.6. A random ﬁxed

point of the ERDS Gψ is a random variable (x , ye ) : Ω → X × Ym , ω → x (ω), ye (ω) on (Ω, F , P) such that

x (ϑω), ye (ϑω) = Gψ 1, ω, x (ω), ye (ω) for all ω ∈ Ω ,

4

Random ﬁxed points can also be deﬁned via a special class of invariant measures, also referred to as random Dirac measures, see Arnold (1998, p. 25).

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4 Economic Models Subject to Stationary Noise

where Ω ⊂ Ω is a ϑ-invariant subset of full measure P(Ω ) = 1.5 A random ﬁxed point (x , ye ) is called asymptotically stable with respect to a norm · on X × Ym if there exists a random neighborhood U(ω) ⊂ X × Ym , ω ∈ Ω such that

lim Gψ t, ω, x0 (ω), y0e (ω) − x (ϑt ω), ye (ϑt ω) = 0 t→∞

for all x0 (ω), y0e (ω) ∈ U(ω) P − a.s. The ﬁrst part in Deﬁnition 4.6 implies

x (ϑt+1 ω), ye (ϑt+1 ω) = Gψ 1, ϑt ω, x (ϑt ω), ye (ϑt ω) for all times t. Hence, a random ﬁxed point generates orbits ,

x (ϑt ω), ye (ϑt ω) t∈N , ω ∈ Ω which solve the random diﬀerence equation e ) = Gψ (1, ϑt ω, xt , yte ), (xt+1 , yt+1

(x0 , y0e ) ∈ X × Ym

induced by (4.5). If Gψ is independent of the perturbation ω, then Deﬁnition 4.6 coincides with the notion of a deterministic ﬁxed point. A random ﬁxed ω ∈Ω point (x , ye ) is asymptotically stable if for almost all perturbations all orbits starting at suﬃciently close points x0 (ω), y0e (ω) ∈ U(ω) eventually converge to orbits

of the random ﬁxed point. By stationarity and ergodicity of ϑ, the process { x (ϑt ·), ye (ϑt ·) }t∈N is stationary and ergodic. Let P denote the probability distribution of (x , ye ). If, in addition, E [ (x , ye ) ] < ∞, then the ergodicity and the stability property of (x , ye ) imply that lim 1 T →∞ T +1 1 T →∞ T +1

= lim

T

χB Gψ (t, ω, x0 (ω), y0e (ω))

t=0 T

χB x (ϑt ω), ye (ϑt ω)

t=0

= P (B)

for all x0 (ω), y0e (ω) ∈ U(ω) P-a.s. In other words, the empirical law of any ‘nearby’ orbit {Gψ t, ω, x0 (ω), y0e (ω) }t∈N is for P-almost all ω ∈ Ω well deﬁned and converges to the true probability distribution of (x , ye ). We are now ready to establish the following notion. 5

In the context of random dynamical systems, the term almost surely (a.s.) is used in a non-standard sense: a property holds a.s. if there exists a ϑ-invariant set Ω ⊂ Ω (ϑΩ = Ω ) such that the property holds for all ω ∈ Ω .

4.4 Random Fixed Points

77

Deﬁnition 4.7. Given an economic law G, a forecasting rule ψ is called asymptotically ε-unbiased if there exists an asymptotically stable random ﬁxed point (x , ye ) of Gψ such that ψ is locally ε-unbiased along (x , ye ), i.e.,

ε (ω) P − a.s. x (ω), ye (ω) ∈ WG For ε = 0, ψ is called asymptotically unbiased. Let ψ be an asymptotically unbiased forecasting rule which is locally unbiased along a random ﬁxed point (x , ye ) of Gψ . Since X = X × Y, we can split x into x = (x , y ) : Ω → X × Y. The forecast errors along (x , ye ) are

(ω) = y (ϑt+1 ω) − ψ ϑt ω, x (ϑt ω), ye (ϑt−1 ω) , ω ∈ Ω, t ∈ N ζt+1 ] = 0 for all times t ∈ N. That is, all mean forecast errors and satisfy Et [ζt+1 conditional on the maximum amount of information available prior to actual realizations vanish. This implies in particular that all forecast errors are zero in the mean, E0 [ζt ] = 0 for all t ∈ N. It follows from the stability property and the ergodicity of (x , ye ) that the sample mean errors and all sample autocorrelation coeﬃcients of all suﬃciently close orbits {xt , yte }t∈N as well as those of the corresponding forecast errors {ζt }t∈N converge almost surely to the respective true means and the true autocorrelation coeﬃcients associated with the random ﬁxed point (x , ye ).

Remark 4.4. The observation just made illustrates that an asymptotically unbiased forecasting rule induces a (stochastic) consistent expectations equilibrium (CEE) in the sense of Hommes & Sorger (1998) and Hommes, Sorger & Wagner (2004), provided that agents in the model have a perceived law of motion whose hypothesized autocorrelation structure is conﬁrmed by the autocorrelation structure of the realizations. Notice, however, that an unbiased forecasting rule is more precise than a forecasting rule which induces a CEE because the former provides the best least-squares prediction given the history of the process. Moreover, the autocorrelation structure of the forecast errors associated with an unbiased forecasting rule may well be non-trivial, cf. B¨ ohm & Wenzelburger (2002) for a concrete example. Hence, learning schemes designed to ﬁnd a CEE such as consistent adaptive learning schemes in the sense of Sch¨ onhofer (1999, 2001) might only be meaningful in situations with additive iid noise. We conclude this section by reviewing a simpliﬁed version of an existence theorem for random ﬁxed points due to Schmalfuß (1996, 1998). Let H(ω) ⊂ Rd , ω ∈ Ω be a random closed set, i.e., each H(ω) is closed a.s. and {ω ∈ Ω | H(ω)∩U = ∅} is measurable for all open sets U ⊂ Rd (Arnold 1998, Prop. 1.6.2, p. 32). Consider random variables h : Ω −→ Rd which are tempered, meaning that

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4 Economic Models Subject to Stationary Noise

lim e−δt h(ϑt ω) = 0 for all δ > 0.

t→∞

Let

, H := h : Ω −→ Rd+mdy | h is tempered and h(ω) ∈ H(ω), ω ∈ Ω

denote the set of all tempered random variables with values h(ω) in H(ω). Theorem 4.3. Let Gψ : Z × Ω × X × Y m → X × Ym be a random dynamical system over a metric dynamical system (Ω, F , P, {ϑt }t∈Z ) as deﬁned in (4.6). Suppose there exists a random set H(ω) ⊂ X × Ym , ω ∈ Ω, such that H is nonempty and the following conditions holds:

(i) The map ω −→ Gψ 1, ϑ−1 ω, h(ϑ−1 ω) is contained in H for all h ∈ H. (ii) There exists a random variable c with E[c] < 0, such that ˜, y˜e )

Gψ (1, ω, x, y e ) − Gψ (1, ω, x ≤ c(ω), ω ∈ Ω. sup log

(x, y e ) − (˜ x, y˜e )

(x,y e ) =(˜ x,˜ y e )∈H(ω)

(iii) If, for some h ∈ H, {Gψ t, ϑ−t ω, h(ϑ−t ω) }t∈N is a Cauchy sequence for each ω ∈ Ω, then its limit is contained in H. Then there exists a unique random ﬁxed point h ∈ H of Gψ with

lim Gψ t, ω, h(ω) − h (ϑt ω) = 0 for all h ∈ H P − a.s. t→∞

If, in addition, the time-one map (x, y e ) → Gψ (1, ω, x, y e ) is continuously diﬀerentiable a.s., then Condition (ii) can be replaced by (ii’) There exists random variable c with E[c] < 0, such that sup (x,y e )∈H(ω)

log D(x,ye ) Gψ (1, ω, x, y e ) ≤ c(ω), ω ∈ Ω

with D(x,ye ) denoting the derivative with respect to (x, y e ). The concept of an asymptotically stable random ﬁxed point in Deﬁnition 4.6 is a natural extension of the notion of an asymptotically stable ﬁxed point in the deterministic case. Combined with the property of unbiasedness, these criteria seem to be natural stability requirements for an ERDS. On a purely distributional level, a simpliﬁed version of Theorem 4.3 is given as Theorem 12.7.2 on p. 430 of Lasota & Mackey (1994). Theorem 4.3 improves the purely statistical viewpoint of considering invariant distributions and Markov equilibria, as done in most economic applications, e.g., see Duﬃe, Geanakoplos, Mas-Colell & McLennan (1994), Bhattacharya & Majumdar (2001), Spear (1988), or Spear & Srivastava (1986). In fact, the required conditions can be established for a number of economic models, yielding a characterization of the long-run behavior of orbits in models with noise. See Chapters 6 and 7 or B¨ ohm & Wenzelburger (2002) for economic examples. A discussion of the relationship between random ﬁxed points and Markov equilibria is found in Schenk-Hopp´e & Schmalfuß (1998).

4.5 Mathematical Appendix

79

Concluding Remarks Extending and conﬁrming results of Chapter 2 to the case of stationary noise, it was shown that the class of unbiased forecasting rules which generate rational expectations are themselves time-invariant functions. The most remarkable phenomenon in stationary economic systems with expectational leads is the notion of an unbiased no-updating forecasting rule which yields the most precise forecasts in the sense that forecast errors vanish conditional on information which was not available at the stage in which they were issued. The existence of unbiased forecasting rules was reduced to the existence of a global inverse of the mean error function associated with the system. This error function depends exclusively on the fundamentals of the economy and is independent of any expectations formation procedure. The information necessary to construct an unbiased forecasting rule requires detailed knowledge of the whole economic system and amounts to the ability of computing the global inverse of the mean error function. The static nature of this error function, however, opens up the possibility to estimate and approximate unbiased forecasting rules from historical data, whenever they exist.

4.5 Mathematical Appendix A random dynamical system in the sense of Arnold (1998) consists of two basic ingredients, a model of the exogenous noise and a model of the system which is perturbed by noise. In what follows we restrict ourselves to the case of discrete time. The noise will be modeled by an (ergodic) metric dynamical system (Ω, F , P, {ϑt }t∈Z ) in the sense of ergodic theory as deﬁned in Assumption 4.1. Recall that any stationary ergodic process can be represented by an ergodic dynamical system. Let X ⊂ Rd be given and B(X) be the σ-algebra of all Borel sets. A random dynamical system (RDS) on X over a metric dynamical system (Ω, F , P, {ϑt }t∈Z ) is a mapping φ : Z × Ω × X → X,

(t, ω, x) → φ(t, ω, x)

with the following properties: (i) Each φ(t, ·), t ∈ Z is F ⊗ B(X)/B(X) measurable. (ii) The mappings φ(t, ω, ·) : X → X form a cocycle over {ϑt }t∈Z , that is, they satisfy φ(0, ω, x) = x for all x ∈ X, ω ∈ Ω and

φ(t1 + t2 , ω, x) = φ t1 , ϑt2 ω, φ(t2 , ω, x)

for all x ∈ X, ω ∈ Ω, and t1 , t2 ∈ Z.

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4 Economic Models Subject to Stationary Noise

If, in addition, each map φ(1, ω, ·) : X → X, ω ∈ Ω, is continuous or even of class C k , then φ is said to be continuous or of class C k , respectively. If the system started at x0 ∈ X, then xt = φ(t, ω, x0 ) describes the state of the system at time t. For an arbitrary initial condition x0 ∈ X and any perturbation ω ∈ Ω, the sequence of points γ(x0 ) := {xt }t∈N with xt = φ(t, ω, x0 ), t ∈ N, is called an orbit of the random dynamical system φ. Having deﬁned a random dynamical system, one may now deﬁne the past and the future associated with such systems. The following deﬁnition is taken from Arnold (1998, Def. 1.7.1, p. 37). Deﬁnition 4.8. A sub-σ-algebra F − of F is called past of the RDS φ if (i) each map φ(t, ·, x), x ∈ X, is F − /B(X)-measurable for all t ≤ 0, and (ii) ϑ−t F − ⊂ F − for all t ≤ 0. A sub-σ-algebra F + of F is called future of the RDS φ if (i) each map φ(t, ·, x), x ∈ X, is F + /B(X)-measurable for all t ≥ 0, and (ii) ϑ−t F + ⊂ F + for all t ≥ 0. The smallest possible choice for the past of φ is F − := σ(φ(s, ·, x)| s ≤ 0, x ∈ X), and the smallest possible choice for the future of φ is F + := σ(φ(s, ·, x)| s ≥ 0, x ∈ X). see Arnold (1998, Sec. 1.7, p. 37). Setting for each t ∈ Z, Ft− := ϑ−t F − and Ft+ := ϑ−t F + , it is clear that Ft− is increasing and Ft+ is decreasing in t ∈ Z, respectively. Consider now the sub-σ-algebras Ft := σ(φ(s, ·, x)| s ≤ t, x ∈ X) and

Ft := σ(φ(s, ·, x)| s ≥ t, x ∈ X)

of F and for each r ∈ N, set . Ft,r = Ft ∩ Ft−r

The sub-σ-algebras Ft and Ft,r have been used in the main text of this chapter, as these seem to be intuitively easier to comprehend. Lemma 4.1. Let φ be a RDS over a metric dynamical system (Ω, F , P, {ϑt }t∈Z ). Then, with the notation introduced above, the following holds for each t ∈ Z: (i) Ft = Ft− ,

(ii) Ft = Ft+ ,

(iii) Ft+s,r = ϑ−s Ft,r , r ∈ N, s ∈ Z.

4.5 Mathematical Appendix

81

Proof. (i) Step 1. We show that Ft− ⊂ Ft . Let s ≤ 0, y ∈ X, and B ∈ B(X) be arbitrary but ﬁxed and consider a typical set Fy− := {ω ∈ Ω| φ(s, ϑt ω, y) ∈ B} ∈ Ft− . To show that Fy− ∈ Ft , let Fx := {ω ∈ Ω| φ(t, ω, x) = y and φ(t + s, ω, x) ∈ B},

x ∈ X.

Since s ≤ 0, Fx ∈ Ft . The cocycle property of φ reads

φ(t + s, ω, x) = φ s, ϑt ω, φ(t, ω, x) , x ∈ X. / Hence x∈X Fx ⊂ Fy− . On the other hand, let ω ∈ Fy− be arbitrary. Since φ(t, ω, ·) is invertible, there exists a unique xω ∈ X such that φ(t, ω, xω ) = y. Hence φ(t + / s, ω, xω ) = φ(s, ϑt ω, y) ∈ B,/implying that ω ∈ Fxω . This shows − that Fy ⊂ x∈X Fx and hence Fy− = x∈X Fx . Since s ≤ 0, y ∈ X, and B ∈ B(X) were arbitrary, this proves Ft− ⊂ Ft . Step 2. The proof is complete if one can show that Ft ⊂ Ft− . To do so, let s ≤ t, x ∈ X, and B ∈ B(X) be arbitrary but ﬁxed. Consider a typical set Fx := {ω ∈ Ω| φ(s, ω, x) ∈ B} ∈ Ft . The cocycle property of φ implies

φ(s − t, ϑt ω, y) = φ s, ω, φ(−t, ϑt ω, y) ,

y ∈ X.

Consider now sets of the form (s ≤ t ﬁxed as above) Fy− := {ω ∈ Ω| φ(−t, ϑt ω, y) = x and φ(s − t, ϑt ω, y) ∈ B} ∈ Ft− , y ∈ X. / By a reasoning analogous to Step 1, Fx = y∈X Fy− which yields Ft ⊂ Ft− . (ii) The proof of Property (ii) is analogous to that of Property (i). (iii) Using (i) and (ii), one has − + ϑ−s Ft,r = ϑ−s Ft− ∩ ϑ−s Ft+ = Ft+s ∩ Ft+s = Ft+s,r .

We use Lemma 4.1 to prove the following Proposition. Proposition 4.1. Let φ be a RDS over a metric dynamical system (Ω, F, P, {ϑt }t∈Z ) as deﬁned above and h : X → Rd be an integrable map. Let Ft = ϑ−t F 0 , t ∈ Z for some sub-σ-algebra F 0 ⊂ F. Then, with the notation introduced above, for each x0 ∈ X,

E h φ(t + 1, ·, x0 ) Ft (ω) = E h φ(1, ϑt ·, xt ) Ft (ω)

= E0 h φ(1, ·, xt ) F 0 (ϑt ω) P-a.s., where xt = φ(t, ω, x0 ).

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4 Economic Models Subject to Stationary Noise

Proof. The assertion follows from Lemma 4.2 and Lemma 4.3 below.

Lemma 4.2. Let (Ω, F , P, {ϑt }t∈Z ) be a metric dynamical system, Ft = ϑ−t F 0 , t ∈ Z for some sub-σ-algebra F 0 ⊂ F, and u : Ω → X be a F/B(X)measurable random variable. Then E[u◦ϑt |Ft ] = E[u|F 0 ]◦ϑt

P − a.s.

Proof. By the deﬁnition of a metric dynamical system, ϑ is measure preserving with respect to P. This implies ϑt P = P with ϑt P denoting the image measure under ϑt . The change-of-variable formula then gives 0 0 0 u d(ϑt P) = u ◦ ϑt dP, F ∈ F 0 . (4.36) u dP = F

ϑ−t (F )

F

On the other hand, the deﬁnition of the conditional expectations and the change-of-variable formula imply 0 0 0 0 udP = E[u|F ] dP = E[u|F 0 ]◦ϑt dP, F ∈ F 0 P−a.s. (4.37) F

F

ϑ−t (F )

Noting that Ft = ϑ−t F 0 and applying the deﬁnition of the conditional expectations again, (4.36) and (4.37) yield 0 0 t E[u ◦ ϑ |Ft ] dP = E[u|F 0 ] ◦ ϑt dP, F ∈ F 0 P − a.s. ϑ−t (F )

ϑ−t (F )

This completes the proof.

Lemma 4.3. Let (Ω, F, P) a probability space, Φ : Rd1 × Rd2 → Rd3 be an integrable map, and x : Ω → Rd1 and y : Ω → Rd2 be integrable random variables. Then

E[Φ(x, y)|y](ω) = E Φ x, y(ω) |y (ω), P − a.s. Here, both conditional expectations are taken with respect to the sub-σ-algebra σ(y) and one is the conditional expectation of the random variable

the second ω → Φ x(ω), η with arbitrary but ﬁxed η ∈ Rd2 . Proof. The factorization lemma (Bauer 1992, p. 71) ensures the existence of a σ(y)-measurable map h such that

E[Φ(x, y)|y](ω) = h y(ω) , P − a.s. and Proposition 15.9 in Bauer (1991, p. 128) shows that 0 0 1 1 h(η) = Φ(x, y)dP = Φ(x, η)dP P(y = η) y−1 (η) P(y = η) y−1 (η)

P − a.s.,

4.5 Mathematical Appendix

83

provided that P(y = η) > 0. Similarly, for each ﬁxed ζ ∈ Rd2 , there exists a σ(y)-measurable map H(·, ζ) such that

E[Φ(x, ζ)|y](ω) = H y(ω), ζ , P − a.s. 0 1 Φ(x, ζ)dP, P − a.s. P(y = η) y−1 (η)

Hence, h y(ω) = H y(ω), y(ω) which completes the proof.

and

H(η, ζ) =

5 Nonparametric Adaptive Learning

In this chapter we introduce a learning scheme for unbiased forecasting rules of the non-linear system (2.8) as introduced in Chapter 2, using nonparametric methods developed in Chen & White (1996, 1998, 2002). Based on results of Yin & Zhu (1990), Yin (1992), and many other authors, these contributions generalize an approach in stochastic approximation theory initiated by Kushner & Clark (1978) to inﬁnite-dimensional spaces.We will specialize these results to economic laws with expectational leads. The main diﬀerence between the learning scheme proposed in this chapter and the approach of the traditional learning literature is that we distinguish between the estimation of an unbiased forecasting rule associated with an economic law and the actual application of a forecasting rule within the system. As in the linear case of Chapter 3, forecasts will be treated systematically as exogenous inputs in the control theoretical sense. One advantage of such an approach is that, contrary to Chen & White (1998), the unknown functional relationship which has be to estimated is uniquely determined. This considerably reduces the mathematical equipment needed to establish convergence results. It allows to select between multiple unbiased forecasting rules on economic grounds and, in principle, it enables a forecasting agency to stabilize the evolving system lat all stages of the learning procedure. The general idea of the learning scheme is to construct an approximation of an unbiased forecasting rule from historical data. A forecasting rule will be updated at each point in time, where we will assume that this rule uses a ﬁnite amount of historical data only. Although a forecasting rule of the general form (2.13) could be taken, our existence and uniqueness results (Theorems 4.1 and 4.2) of Chapter 4 suggest to take no-updating rules. Moreover, as Theorem 2.1 of Chapter 2 showed, unbiased no-updating rules provide the most precise forecasts. Assuming existence and uniqueness, we will show how to approximate an unbiased no-updating forecasting rule from an estimate of the ‘inverse’ of the economic law. If it is unknown whether or not such a forecasting rule exists, then the economic law G should be directly estimated. As all multiplicities

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5 Nonparametric Adaptive Learning

are embodied in the economic law, an approximated unbiased forecasting rule could then be computed from an estimated economic law. The estimation techniques are completely analogous. We start with an outline of the general methodology.

5.1 The Method of Stochastic Approximation Recall that, given the information vector of past observed states e e Zt = (yt−1 , xt−1 , . . . , yt−r , xt−r ) ∈ Σ e e in an arbitrary period t, the vector of forecasts yte = (yt,t+1 , . . . , yt,t+m ) formed (1) in period t by means of a no-updating forecasting rule Ψ = (Ψ , . . . , Ψ (m) ) is e e yt,t+i = Ψ (i) (xt , Zt ) := yt−1,t+i , i = 1, . . . , m − 1, (5.1) e yt,t+m = Ψ (m) (xt , Zt ).

If (xt−1 , Zt−1 ) ∈ X × Σ ⊂ RdH denotes the state of the system in period t − 1, then in view of Assumption 2.1 ⎧

e xt = G ξt (ω), xt−1 , yt−1 ) ⎪ ⎪ ⎨ e , xt−1 , prΣ Zt = (yt−1 (5.2) −r Zt−1 ⎪ ⎪ ⎩ e yt = Ψ (xt , Zt ) describes the state of the system in period t under the perturbation ω ∈ Ω, where as before prΣ −r

:Σ→

r−1 &

(Ym × X),

(Z (1) , . . . , Z (r) ) → (Z (1) , . . . , Z (r−1) )

(5.3)

i=1

denotes the projection into the ﬁrst r − 1 components. Remark 5.1. The information contained in Zt is redundant when using noupdating forecasting rules of the form (5.1). In this case e e yt,t−j+m = yt−j,t−j+m ,

j = 1, . . . , m − 1,

e e , . . . , yt,t+m ) are redundant so that the ﬁrst m − 1 components of yte = (yt,t+1 if r > m.

In order to formulate a learning scheme according to which the forecasting rule Ψ is updated, we need to specify the set of feasible forecasting rules from which an agent or a forecasting agency can choose. Let X × Σ be an open and convex subset of RdH and · denote the Euclidean norm. Restricting attention to no-updating forecasting rules, it suﬃces to consider the function Ψ (m) for

5.1 The Method of Stochastic Approximation

87

p the m-th forecast of a forecasting rule Ψ . Let CB (X×Σ; Rdy ) denote the space of all bounded p-times continuously diﬀerentiable functions ψ : X × Σ → Rdy with bounded derivatives and CB (X × Σ; Rdy ) be the space of all functions p (X × Σ; Rdy ) is which are continuous and bounded.1 It is well known when CB a dense subspace of a real separable Hilbert space H. This could be either the space of all square integrable functions L2 (X × Σ; Rdy ) or the Sobolev space Wp,2 (X × Σ; Rdy ).2 We will choose the Sobolev spaces H := Wp,2 (X × Σ; Rdy ) and H := p,2 W (X × Σ; Rmdy ) with p > d2H as the underlying Hilbert spaces of our learning scheme. By the Sobolev Imbedding Theorem (see Theorem 5.4 below) which will be applied in the sequel, H can be imbedded into CB (X × Σ; Rdy ) in the sense that if ψ ∈ H, then there exists ψ0 ∈ CB (X × Σ; Rdy ) such that ψ = ψ0 almost everywhere. The Sobolev Imbedding Theorem requires that X×Σ has a geometric property, called the cone property. Open balls and open rectangular domains are examples for sets having the cone property, so that from an economic view point this assumption imposes no serious restriction. A brief discussion of all necessary details is found in Section 5.5.1 below. The Hilbert space H is endowed with an inner product and inner-product induced norm, denoted by ·, ·H and · H, respectively. The σ-algebra generated by the subsets of H which are open in the norm-induced topology of H is denoted by B(H). Similarly, we denote by H the Hilbert space of all functions Ψ : X × Σ → Rmdy endowed with the norm · H . Except for the dimension of the target space Rmdy , we assume throughout that H and H are Hilbert spaces of the same sort. Fix p > d2H as above and set p S := CB (X × Σ; Ym )

(5.4)

for the space of feasible forecasting rules. Note that any feasible forecasting rule Ψ must take values in Ym and that S is a subspace of H . Let πY denote a projection of Rdy onto Y. Deﬁne an imbedding I of H into H by I : H → H ,

ψ → I(ψ),

such that I(ψ) : X × Σ → Ym is given by

I(ψ) x, Z) := (z (2) , . . . , z (m) , πYm ◦ ψ(x, Z) ,

(5.5)

(x, Z) ∈ X × Σ,

where Z = (z (1) , . . . , z (m) , Z ) ∈ Σ and (z (1) , . . . , z (m) ) ∈ Ym . In particular, p (X × Σ; Rdy ) into a feasible forecasting rule I(ψ) ∈ I transforms any ψ ∈ CB p m CB (X × Σ; Y ). 1

Throughout this chapter we change the notation with respect to Chapter 4 and denote the forecasting function for the m-th forecast by ψ for ease of notation. 2 As an alternative, the space of all p-times continuously diﬀerentiable functions with compact support could be chosen. By Theorem 3.18, p. 54 of Adams (1975), this space is a dense subset of a Sobolev space as well if X × Σ has an additional geometric property, called the segment property.

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5 Nonparametric Adaptive Learning

With these choices, the learning scheme which will be introduced below may require that the forecasting rule obtained from an estimation procedure can be censored. The main reason for such a manipulation is to generate certain stochastic properties of the estimates that guarantee convergence of the algorithm. To this end, we introduce for each t ∈ N a censor map Υt : H → H ,

ψ → Υt (ψ)

(5.6)

p which maps CB (X × Σ; Rdy ) onto S. The simplest example is Υt ≡ I for all t ∈ N when no censoring is applied. We next deﬁne the way in which a forecasting rule will be updated. (m) the m-th component funcThroughout this chapter, denote by ψ = Ψ tion of an unbiased no-updating Ψ introduced in Deﬁnition 2.3 of Section 2.5 in Chapter 2. Clearly, Ψ = I(ψ ). Let {at }t∈N be a sequence of positive real numbers, which will be speciﬁed later in this section, and for each t ∈ N, let

Mt : X × Σ × H −→ H be a function that governs the updating of a forecasting rule at time t ∈ N. Assume that each Mt (x, Z, ·), t ∈ N, (x, Z) ∈ X × Σ maps diﬀerentiable functions onto diﬀerentiable functions. The resulting learning scheme then evolves over time according to the list of mappings

⎧ e xt = G ξt (ω), xt−1 , yt−1 ⎪ ⎪ ⎪

e ⎪ ⎪ ⎪ Zt = yt−1 , xt−1 , prΣ ⎪ −r Zt−1 ⎪ ⎨ yte = Ψt (xt , Zt ) (5.7) ⎪ ⎪ ⎪ ⎪ Ψt = Υt (ψˆt ) ⎪ ⎪ ⎪ ⎪ ⎩ˆ ψt = ψˆt−1 + at−1 Mt−1 (xt , Zt , ψˆt−1 ) with t ∈ N. The ﬁrst three equations in the recursive system (5.7) are already given in (5.2) and play the role of the former time-one map (2.15) [or, equivalently, (4.5)]. ψˆt is a period-t approximation of the m-th component ψ of the desired unbiased no-updating forecasting rule Ψ = I(ψ ). The censored map Ψt = Υt (ψˆt ) is the forecasting rule which is applied in period t. As mentioned above this may become necessary in order to fulﬁll certain stochastic requirements for the estimation technique and in order to stabilize the evolving system (5.7). The ﬁfth equation in (5.7) speciﬁes the way in which the estimate ψˆt is updated. This updating scheme is commonly referred to as a Robbins-Monro procedure with feedback. The basic idea of the algorithm (5.7) is to ﬁnd the unknown function ψ as follows. Suppose that ψ can be characterized as the zero of an unknown map M : H → H. Assume that the unknown map M is approximated by a sequence of functions Mt , t ∈ N. Then ψ could recursively be estimated by taking some initial element ψˆ0 ∈ H and setting

5.1 The Method of Stochastic Approximation

ψˆt+1 = ψˆt + at Mt (xt+1 , Zt+1 , ψˆt )

= ψˆt + at Ut (xt+1 , Zt+1 , ψˆt ) + M (ψˆt ) ,

89

t = 0, 1, . . . ,

where Ut (xt+1 , Zt+1 , ψˆt ) := Mt (xt+1 , Zt+1 , ψˆt ) − M (ψˆt ) is the approximation error of period t. We will show that ψˆt − ψ H converges almost surely to zero as t -tends to inﬁnity, provided that the approximation errors , Ut (xt+1 , Zt+1 , ψˆt ) t∈N fulﬁll certain stochastic requirements. In order to establish such convergence results, the magnitudes ψˆt H of the estimates ψˆt must be hindered from diverging. This will be achieved by modifying the learning scheme (5.7) as follows. The idea is to ‘reset’ the estimate ψˆt to some ﬁxed forecasting rule ψ ∈ H whenever the sequence {ψˆt }t∈N attempts to escape. To this end, let χJ denote the indicator function of a subset J ⊂ Ω and {bt }t∈N be a sequence of strictly increasing positive real numbers. Deﬁne a sequence of positive integer-valued random variables {γ(t)}t∈N by γ(0) = 0, γ(t + 1) = γ(t) + χJtc , (5.8) where

# " Jt := ψˆt−1 + at−1 Mt−1 (xt , Zt , ψˆt−1 ) H ≤ bγ(t) , " # Jtc := ψˆt−1 + at−1 Mt−1 (xt , Zt , ψˆt−1 ) H > bγ(t) .

Fix some ¯b with 0 < ¯b < b1 and choose a ﬁxed ψ ∈ H with ψ H < ¯b. For each t ∈ N, set

⎧ e xt = G ξt (ω), xt−1 , yt−1 ⎪ ⎪ ⎪

e ⎪ ⎪ ⎪ , xt−1 , prΣ ⎪ Zt = yt−1 −r Zt−1 ⎪ ⎪ ⎨ e yt = Ψt (xt , Zt ) (5.9) ⎪ ⎪ ⎪ ˆ ⎪ Ψt = Υt (ψt ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ψˆt = ψˆt−1 + at−1 Mt−1 (xt , Zt , ψˆt−1 ) χJ + ψ χJ c t t Both systems (5.7) and (5.9) are variants of learning schemes developed in Chen & White (1998, 2002). The main diﬀerence is that we distinguish between the recursive approximation of a forecasting rule and its application to the system such that the forecasts which actually feed back into the process may be altered or, in other words, censored. The learning scheme (5.9) is also referred to as randomly truncated Robbins-Monro procedure with feedback cf. Chen & White (1998). Remark 5.2. In the case in which {ξt }t∈N is iid, Remark 2.4 of Chapter 2 showed that ψ is a function of the form ψ : X × Ym−1 → Y. In this case, Σ = Ym , r = 0 and one may set

90

5 Nonparametric Adaptive Learning p S := CB (X × Ym−1 ; Ym )

for the space of feasible forecasting rules. The Hilbert spaces H and H are deﬁned accordingly. The same remark applies to the case of stationary ergodic noise with r = 0 treated in Chapter 4. Remark 5.3. The stochastic approximation schemes in (5.7) and (5.9) involve a generalization of a stochastic approximation method developed by Kushner & Clark (1978) to inﬁnite-dimensional systems with feedback, see also Kushner & Yin (2003) and references therein. The original algorithm without an expectations feedback dates back to Robbins & Monro (1951). Kushner & Clark (1978) extended this method to ﬁnite-dimensional systems with feedback, Yin & Zhu (1990), Yin (1992), and others developed a generalization to systems with an inﬁnite-dimensional state space. Chen & White (1996, 1998, 2002) extended and reﬁned these results to feedback systems with an inﬁnite-dimensional state space. For an account of the literature we refer to Chen and White’s as well as to Yin and Zhu’s work. Remark 5.4. The system (5.9) is not necessarily a random dynamical system in the sense of Arnold (1998) because the fourth and ﬁfth equation in (5.9) will most generally violate the cocycle property (see Chapter 4).

5.2 A Basic Convergence Result We are now in a position to formulate a convergence result which is based on Yin & Zhu (1990) and Chen & White (1996, 2002). The convergence result stated in Theorem 5.1 below is quite general, whereas the required assumptions are rather abstract. Indeed, the crucial assumption which guarantees convergence will, in general, be diﬃcult if not impossible to verify. A list of assumptions which is easier to verify will be formulated in Section 5.4. There are various approaches using diﬀerent assumptions to prove almost sure convergence of the sequence of estimates {ψˆt }t∈N to ψ . In essence, all try to establish the following two relations: 1. Boundedness of the sequence of estimates, that is, sup ψˆt H < ∞ P − a.s. t∈N

2. For each T > 0, convergence of the sum lim

sup

'n−1 ' ' ' aj Mj (xj+1 , Zj+1 , ψˆj ) − M (ψˆj ) ' = 0 ' H

t→∞ t
where for each t ∈ N, i−1 " # n(t, T ) := max i > t aj ≤ T . j=t

P − a.s.,

5.2 A Basic Convergence Result

91

Before stating our assumptions, let us introduce some notation. Given some feasible forecasting rule Ψ ∈ S and an economic law G of the form (2.8), it will turn out to be convenient to allow the time-one map as deﬁned in (2.15) to depend on Ψ . Formally, we deﬁne ⎧ ⎪ ⎨Ξ × X × Σ × S −→ X × Σ

. (5.10) G: G ξ, x, Ψ (x, Z) ⎪

−→ ⎩ (ξ, x, Z, Ψ ) Σ Ψ (x, Z), x, pr−r Z The map (5.10) will replace the role of the former time-one map (2.15) as 1t+1 follows. Let i=1 S denote the (t + 1)-fold product of S and Ψ0t = (Ψ0 , . . . , Ψt ) ∈

t+1 &

S

i=1

be a list of forecasting rules, where Ψs is the forecasting rule applied in period s. Deﬁne maps G(t, ·) : Ω × X × Σ ×

t &

S → X × Σ,

t∈N

(5.11)

i=1

recursively by setting

G t, ω, x0 , Z0 , Ψ0t−1

if t > 0, G ξt (ω), G t − 1, ω, x0 , Z0 , Ψ0t−2 , Ψt−1 := (x0 , Z0 ) if t = 0. If the system started in some state (x0 , Z0 ) ∈ X × Σ, then the t-th iteration of the map (5.10) under the perturbation ω ∈ Ω induces maps (5.11) such that

(xt , Zt ) = G t, ω, x0 , Z0 , Ψ0t−1 describes the state of the system at time t. If Ψ0t−1 = (Ψ, . . . , Ψ ) is a constant list of functions so that Ψs = Ψ for s = 0, . . . , t − 1, we write

(5.12) G (t, ω, x0 , Z0 , Ψ ) = G t, ω, x0 , Z0 , Ψ0t−1 for notational convenience. The following assumptions are taken from Yin & Zhu (1990) and Chen & White (2002) and adjusted to our setup. The ﬁrst assumption is a slight extension of our previous Assumption 2.1. The second one concerns the dynamic stability of our system under learning. Recall that in the context of inﬁnite dimensional spaces, a mapping is said to be bounded if it maps bounded sets into bounded sets, where we mean bounded with respect to the norm topology.

92

5 Nonparametric Adaptive Learning

Assumption 5.1. The exogenous noise is driven by a sequence of random variables {ξt }t∈N on a complete probability space (Ω, F , P) with values in a subset Ξ ⊂ Rdξ , such that for each t ∈ N, ξt : Ω → Ξ. Assumption 5.2. (i) X×Σ ⊂ RdH is open, convex and has the cone property. (ii) Each map Υt : H → S ⊂ H , t ∈ N is uniformly continuous on bounded sets. That is, for each bounded subset A ⊂ H and each > 0, there exists some δ = δ(, A) such that ˜ H < δ implies Υt (ψ) − Υt (ψ)

˜ H <

ψ − ψ

for all ψ, ψ˜ ∈ A. (iii) The economic law G is a continuous map and there exists a compact subset D ⊂ X × Σ which is forward-invariant in the sense that

G ·, Υt (·) : Ξ × D × H → D, uniformly in t ∈ N. The boundedness condition in Assumption 5.2 imposes an important stability condition on the recursive system (5.9). This sets aside the problem that forecasting rules could well destabilize the system in the sense that the endogenous state variables {(xt , Zt )}t∈N generated by the Robbins-Monro procedure (5.9) diverge. This possibility constitutes a serious control problem. It is ruled out by imposing the strong assumption that a list of censoring maps {Υt }t∈N is known which keeps the system stable no matter what the actual estimate for a forecasting rule is. It is beyond the scope of these notes to analyze the case in which the unknown system has to be stabilized without this a-priori knowledge. However, in principle, stability could be achieved through a reset mechanism similar to the one in (5.9). Notice in this context that the forecasting rule ψ would have to be known so that the system G is stable when ψ is applied in the sense that a bounded forward-invariant set exists. Example 5.1. As an example for a time-invariant censor map, consider fore1 (X × Σ; Rdy ) with bounded ﬁrst derivative. Let Dψ(x, Z) casting rules ψ ∈ CB denote the derivative (i.e., the Jacobi matrix) of Ψ at (x, Z) and # "' ' ' ' 'Dψ ' := sup 'Dψ(x, Z)' (x, Z) ∈ X × Σ ∞ be the supremum norm of the derivative. Choose a positive constant 0 < cΥ < 1 and deﬁne 1 1 CB (X × Σ; Rdy ) (X × Σ; Rdy ) −→ CB Υ : (5.13) ψ −→ Υ (ψ) by setting

Υ (ψ) =

ψ cΥ Dψ∞

ψ

if Dψ ∞ < cΥ . if Dψ ∞ ≥ cΥ

5.2 A Basic Convergence Result

93

The function (5.13) maps diﬀerentiable functions onto diﬀerentiable contractions with a prescribed Lipschitz constant cΥ , so that ' ' ' ' 'Υ (ψ)(x, Z) − Υ (ψ)(˜ ˜ ' ˜ ' ≤ cΥ '(x, Z) − (˜ x, Z) x, Z) ˜ ∈ X × Σ. For each t ∈ N, deﬁne for all (x, Z), (˜ x, Z) p Υt ≡ Υ : CB (X × Σ; Rdy ) → S,

ψ → I ◦ Υ (ψ),

(5.14)

where I is deﬁned in (5.5) and S is the space of all feasible forecasting rules. Based on the Sobolev Imbedding Theorem, Lemma 5.6 of Section 5.5.1 below shows that the censor map (5.14) can be continuously extended to a map on H which satisﬁes Assumption 5.2 (ii). Notice, however, that Υ (ψ ) = Ψ only if Dψ ∞ ≤ cΥ . Let us describe now the assumptions concerning the updating of estimates {ψˆt }t∈N . The ﬁrst assumption stipulates the properties of the gains {at }t∈N involved in the algorithm (5.9), also referred to as step sizes. Assumption 5.3. {at }t∈N is a sequence of exogenously given non-increasing positive numbers such that the following conditions hold: !∞ (i) at → 0 as t → ∞, (ii) t=0 at = ∞, (iii) at ≤ at+1 (1 + a at )

for some 0 < a ≤ 1.

A popular choice satisfying Assumption 5.3 is at = 1t , t ∈ N. The updating functions {Mt }t∈N are speciﬁed next. Let {Hk }k∈N be a sequence of closed, bounded, and convex subsets of H. Suppose that Hk ⊂ Hk+1 for all k ∈ N / and that the closure of k∈N Hk with respect to the norm topology is equal to the Hilbert space H. Assumption 5.4. The updating functions {Mt }t∈N take the following form. Let k(·) : N → N, t → k(t) be a non-decreasing function with 1 ≤ k(t) ≤ k(t + 1) ≤ k(t) + 1 For each t ∈ N, let

and

lim k(t) = ∞.

t→∞

Mt : X × Σ × H → Hk(t+1)

be a B(X × Σ × H)/B(H) measurable and bounded map. Example 5.2. Let πk : H → Hk denote a projection operator, i.e. πk ◦πk = πk and M : X × Σ × H → H be a measurable and bounded map. Then Mt := πk(t) ◦ M , t ∈ N satisﬁes the conditions of Assumption 5.4. Such updating functions have been proposed in Chen & White (1998, 2000)

94

5 Nonparametric Adaptive Learning

Assumption 5.4 allows us to introduce nonparametric recursive estimators as kernel, orthogonal series, splines, wavelets, and neural-network estimators, etc., see White, Gallant, Hornik, Stinchcombe & Woolridge (1992) or Prakasa Rao (1983) for examples. Example 5.3. An example for a sequence {Hk(t) }t∈N of closed, bounded, and convex subsets of H can be deﬁned as follows. Recall that dH is the dimension p of X × Σ and let φ ∈ CB (X × Σ; R) be a cumulative distribution function. For each l = 1, . . . , dy , j ∈ N, let each Alj : RdH → R be a linear map and ηlj ∈ RdH be a vector. Choose an increasing sequence of positive real numbers {βt }t∈N such that βt → ∞ as t → ∞. For each t ∈ N, deﬁne " Hk(t) := ψ = (ψ1 , . . . , ψdy ) : X × Σ → Rdy ψl (ζ) =

k(t)

αlj φ(Alj ζ + ηlj ), where αlj ∈ R and

j=1

, max ηlj , Alj ≤ βt , l = 1, . . . , dy

2 .

1≤j≤k(t)

It is shown in / White, Gallant, Hornik, Stinchcombe & Woolridge (1992) that the closure of t∈N Hk(t) is equal to H, whereas Hk(t) ⊂ Hk(t+1) for all t ∈ N. Our next assumption concerns the unknown ‘limiting map’ M : H → H which characterizes the desired unbiased forecasting rule Ψ . Assumption 5.5. There exists a measurable map M : H → H such that the following holds: (i) M has a unique zero ψ ∈ H, that is, M (ψ ) = 0. Ψ = I(ψ ) is the unique unbiased no-updating forecasting rule associated with the economic law G. (ii) M is uniformly continuous on bounded sets. That is, for each bounded subset A ⊂ H and each > 0, there exists δ = δ(, A) such that ˜ H< ˜ H < δ implies M (ψ) − M (ψ)

ψ − ψ

for all ψ, ψ˜ ∈ A. Assumption 5.5 (ii) implies that M is continuous and maps bounded sets into bounded sets. Yin & Zhu (1990) give examples of M that satisfy Assumption 5.5 (ii). These include linear operators, H¨ older and Lipschitz operators, continuous operators with ‘polynomial growth’ and some compact operators.

5.2 A Basic Convergence Result

95

Remark 5.5. The limiting function M can be related to our mean error function EG as given in Deﬁnition 2.2 of Chapter 2. A typical candidate for M could be of the form M (ψ) := [ψ − ψ] f (ψ),

(5.15)

where f (ψ) is a suitably deﬁned R+ -valued function on X × Σ which depends on the forecasting rule ψ. Thus M deﬁnes weighted forecast errors in terms of forecasting rules. The next assumption may be interpreted as a ‘stability’ assumption which is used quite often when treating stochastic approximation problems. Assumption 5.6. There exists a bounded and twice continuously Fr´echet differentiable functional V : H → R, also referred to as Liapunov function. Let V denote the gradient of the Fr´echet derivative3 of V. Suppose that V satisﬁes the following properties: (i) V(ψ ) = 0 and (ii) V(ψ) > 0 (iii) V(ψ) < dV

lim

ψH →∞

V(ψ) = ∞.

V (ψ), M (ψ)H < 0 for all ψ ∈ H, ψ = ψ . := inf{V(ψ) | ψ H > ¯b}.

and

Assumption 5.6 implies that V maps bounded subsets of H into bounded subsets of R. When M is Fr´echet diﬀerentiable at the unique zero ψ , we can choose V to be of a local quadratic form

V(ψ) = (ψ − ψ ), DM (ψ )(ψ − ψ )H + o ψ − ψ 2H , where DM is the Fr´echet derivative of M . In general, however, diﬀerentiability of M will not be imposed. Example 5.4. A typical example for a Liapunov function is the functional V : H → R, deﬁned by V(ψ) := ψ − ψ 2H . It is easy to show that in this case the gradient is V (ψ) = 2[ψ − ψ ]. If, in addition, the limiting function M is given by (5.15), then the ﬁrst two conditions of Assumptions 5.6 are satisﬁed. We are now ready to state our ﬁrst result which assures that the sequence of estimates {ψˆt }t∈N remains bounded with respect to the norm · H on H.

3

If DV(ψ) denotes Fr´echet derivative at ψ ∈ H, then the gradient V is deﬁned

as see Berger (1977).

DV(ψ)(ϕ) = V (ψ), ϕH ,

ψ, ϕ ∈ H,

96

5 Nonparametric Adaptive Learning

Lemma 5.1. Let Assumptions 5.1-5.6 be satisﬁed and (x0 , Z0 ) ∈ D, ψˆ0 ∈ H0 be given. If t ' '

' ' lim sup 'at Mi G ξi+1 (ω), xi , Zi , Υi (ψˆi ) , ψˆi − M (ψˆi ) ' < 1 t→∞

H

i=1

P-a.s., then

lim γ(t) = γmax < ∞

t→∞

P-a.s.

for the integer-valued random variable γ(t) deﬁned in (5.8). Moreover, the learning scheme (5.9) satisﬁes sup ψˆt H ≤ bγmax P − a.s. t∈N

The proof of Lemma 5.1 is that of Proposition 4.1 on p. 124 in Yin & Zhu (1990). By Lemma 5.1 there exists a random integer tmin (ω) ∈ N such that for all t ≥ tmin (ω), ψˆt = ψˆt−1 + at−1 Mt−1 (xt , Zt , ψˆt−1 ). In other words, after some time tmin (ω) the algorithm (5.9) produces no truncations. The critical assumption which will guarantee convergence is given next. Assumption 5.7. Let (x0 , Z0 ) ∈ D and ψˆ0 ∈ H0 be arbitrary with D as given in Assumption 5.2 (iii). Assume that t ' '

' ' lim sup 'at Mi G ξi+1 (ω), xi , Zi , Υi (ψˆi ) , ψˆi − M (ψˆi ) ' = 0 t→∞

i=1

H

P-a.s. It is shown in Theorem 3.3 of Yin & Zhu (1990) that Assumption 5.7 is necessary for almost-sure convergence of the estimates. Kronecker’s Lemma (see Lukacs 1975, Lemma 4.3.3, p. 96) together with Assumption 5.3 implies that Assumption 5.7 is satisﬁed, provided that P-a.s. t ' '

' ' lim sup ' ai Mi G ξi+1 (ω), xi , Zi , Υi (ψˆi ) , ψˆi − M (ψˆi ) ' < ∞ t→∞

i=1

H

for arbitrary (x0 , Z0 ) ∈ D, ψˆ0 ∈ H0 . We are now ready to state our ﬁrst convergence result. Based on Yin & Zhu (1990, Thm. 3.2, p. 123) this result is given as Theorem 2.2 in Chen & White (2002). We present a slightly extended version in which the expectations feedback is incorporated. In the literature, this situation is also referred to as the case with correlated noise.

5.2 A Basic Convergence Result

97

Theorem 5.1. Let Assumptions 5.1-5.7 be satisﬁed and let (x0 , Z0 ) ∈ D and ψˆ0 ∈ H0 be arbitrary initial conditions. If {ψˆt }t∈N is a sequence of functions in H generated by the learning scheme (5.9), then lim ψˆt − ψ H = 0

t→∞

P − a.s.

If, in addition, limt→∞ Υt (ψ ) = Ψ , then lim Ψt − Ψ H = 0

t→∞

P − a.s.

Proof. The ﬁrst part of the proof is that of Theorem 3.1 of Yin & Zhu (1990), see also Theorem 2.2 in Chen & White (2002). The second part follows from

the uniform continuity of Υt , t ∈ N presumed in Assumption 5.2 (ii). Theorem 5.1 shows that the learning scheme (5.9) generates estimates {ψˆt }t∈N which in the strong norm topology on H converge almost surely to the m-th component ψ of the unbiased forecasting rule Ψ = I(ψ ) for all initial conditions ψˆ0 ∈ H0 and thus globally on H. Contrary to Chen & White (1998), we distinguished between the convergence of the estimates ψˆt and the convergence of the respective forecasting rules Ψt = Υt (ψˆt ) which are actually used for forecasting. Notice that the convergence result of Theorem 5.1 does not require that censoring can be removed as time tends to inﬁnity. The diﬃculty with Theorem 5.1 is that in practice it will be hard if not impossible to verify the crucial Assumption 5.7 directly. For this reason conditions which are easier to verify are needed. We split these conditions into assumptions concerning the updating functions and into an assumption concerning the feedback eﬀect of forecasts. Assumption 5.8. The updating functions {Mt }t∈N given in Assumption 5.4 satisfy the following additional conditions. (i) The sequence {Mt }t∈N is bounded on bounded sets uniformly in t. That is, for any bounded subset B ⊂ X × Σ × H, there exists a constant cB such that for each t ∈ N,

Mt (x, Z, ψ) H ≤ cB

for all (x, Z, ψ) ∈ B.

(ii) There exists a sequence {mt }t∈N of B(H)/B(R)-measurable functions mt : H → R+ , t ∈ N, and a constant bH ∈ R+ , such that: ˜ ∈ X × Σ, all ψ ∈ H, and each t ∈ N, (a) for all (x, Z), (˜ x, Z) ' ' ˜ ψ) H ≤ mt (ψ) '(x, Z) − (˜ ˜ ', x, Z, x, Z)

Mt (x, Z, ψ) − Mt (˜ (b) there exists m ∈ R+ , so that for each t ∈ N, , sup mt (ψ) ψ ∈ H with ψ H ≤ bH ≤ m.

98

5 Nonparametric Adaptive Learning

(iii) There exists a sequence {ht }t∈N of B(X × Σ)/B(R)-measurable functions ht : X × Σ → R+ , t ∈ N, and a constant h ∈ R+ , such that: (a) for all (x, Z) ∈ X × Σ, all ψ, ψ˜ ∈ H, and each t ∈ N, ˜ H ≤ ht+1 (x, Z) ψ − ψ

˜ H.

Mt (x, Z, ψ) − Mt (x, Z, ψ)

(b) for each (x0 , Z0 ) ∈ D, with D as given in Assumption 5.2 (iii), and each diﬀerentiable ψ ∈ H with ψ H ≤ bH ,

sup E ht+1 G(ξt+1 (·), xt , Zt , Υt (ψ)) ≤ h t∈N

and lim at

t→∞

t

hj+1 G(ξj+1 (ω), xj , Zj , Υj (ψ)) j=1

−E hj+1 G(ξj+1 (·), xj , Zj , Υj (ψ)) =0

P − a.s.

(iv) For each (x0 , Z0 ) ∈ D and each diﬀerentiable ψ ∈ H with ψ H ≤ bH , t ' ' ' ' Mj (G(ξj+1 (ω), xj , Zj , Υj (ψ)), ψ) − M (ψ) ' = 0 lim 'at

t→∞

H

j=1

P − a.s.

The assumption on the feedback eﬀect of forecasts concerns essentially the eﬀect of the sequence of censor maps {Υt }t∈N on the economic law G and is formalized as follows. Assumption 5.9. Let (x0 , Z0 ) ∈ D be arbitrary. Assume that a sequence of censor maps {Υt }t∈N is given so that for each > 0, there exists δ = δ() such that P-a.s. ' ' 'G(t + 1, ω, x0 , Z0 , Ψ0t ) − G(t + 1, ω, x0 , Z0 , Ψ0 )' < , t ∈ N (5.16) for any sequence of diﬀerentiable forecasting rules Υt (ψˆt ) which satisﬁes

ψˆt − ψˆ0 H < δ

,

ψˆt

t∈N

in H with Ψt =

for all t ∈ N.

Assumptions 5.8 and 5.9 will now be used to establish the result of Lemma 5.1 for the case of an expectations feedback. The following proposition is a variant of Lemma B.1 in Chen & White (1998). Proposition 5.1. Let Assumptions 5.1-5.6 and Assumptions 5.8 and 5.9 be satisﬁed with an arbitrarily large bH . Then for arbitrary (x0 , Z0 ) ∈ X × Σ and ψˆ0 ∈ H0 ,

5.3 Some Probabilistic Prerequisites

lim γ(t) = γmax < ∞

t→∞

99

P-a.s.

for the integer-valued random variable γ(t) deﬁned in (5.8). Moreover, the estimates of the learning scheme (5.9) satisfy sup ψˆt H ≤ bγmax P − a.s. t∈N

and there exists tmin (ω), so that no truncations occur for all t ≥ tmin (ω). Throughout the remainder of this chapter we tacitly assume that bγmax ≤ bH . The following Theorem 5.2 shows that Assumptions 5.8 and 5.9 imply Assumption 5.7, so that convergence of the learning scheme (5.9) obtains. Theorem 5.2. Let Assumptions 5.1-5.6 be satisﬁed. Let (x0 , Z0 ) ∈ D and ψˆ0 ∈ H0 be arbitrary initial conditions and {ψˆt }t∈N be a sequence of functions in H generated by the learning scheme (5.9). If, in addition, Assumptions 5.8 and 5.9 hold, then Assumption 5.7 is satisﬁed and all conclusions of Theorem 5.1 hold. The proofs of Proposition 5.1 and Theorem 5.2 are given in Section 5.5.2 below. In order to provide further conditions which are easier to verify than the ones given so far, we need some probabilistic prerequisites. These will be discussed next.

5.3 Some Probabilistic Prerequisites We brieﬂy summarize some concepts for Banach-space-valued random variables which may be found in Egghe (1984, pp. 1-9) or Yosida (1978). Let (Ω, F , P) be a complete probability space and (B, · B ) denote a real separable Banach space with norm · B . B is endowed with the Borel σ-algebra B(B) generated by the subsets of B which are open in the norm topology. A F /B(B)-measurable map D : Ω → B is called B-valued random variable. The Bochner integral of a Banach-space-valued map is deﬁned as follows. Deﬁnition 5.1. Let D : Ω → B be F /B(B)-measurable. D is Bochner integrable (with respect !nto P) if there exists a sequence {Dn }n∈N of simple functions, i.e. Dn := i=1 hni χAni , where hni ∈ B, Ani ∈ F with Ani ∩ Anj = ∅ for i = j, such that P-a.s. 0 (i) lim D − Dn B = 0, (ii) lim

D(ω) − Dn (ω) B P(dω) = 0. n→∞

n→∞

Ω

The Bochner integral of D is deﬁned as 0 0 D(ω)P(dω) := lim Dn (ω) P(dω). Ω

n→∞

Ω

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5 Nonparametric Adaptive Learning

It can be shown that a B-valued function D is Bochner integrable if and only if D B is Lebesgue integrable, see Yosida (1978, Thm. 1, p. 133). In this case 0 E[D] := D(ω) P(dω) Ω

is ﬁnite and will be referred to as expectation of D. We denote by Lp (Ω; B) ≡ Lp (Ω, F , P; B),

1≤p<∞

the space of (equivalence classes of) functions D : Ω → B which are F /B(B)measurable such that the map D(·) pB is Lebesgue integrable. That is,

D(·) B ∈ Lp (Ω, F , P). For 1 ≤ p < ∞, Lp (Ω; B) becomes a Banach space when endowed with the norm $0 % p1

D(ω) pB P(dω)

D Lp := Ω

and we write D ∈ Lp (Ω; B) if D Lp < ∞. For any sub-σ-algebra A of F , let Lp (Ω, A; B) ≡ Lp (Ω, A, P|A; B),

1≤p<∞

denote the space of all A/B(B)-measurable functions D with D ∈ Lp (Ω; B). By Theorem I.2.2.1 in Egghe (1984, p. I.2) there exists a unique map E[·|A] : L1 (Ω; B) → L1 (Ω, A; B) such that 0 0 E[D|A](ω) P(dω) = D(ω) P(dω) for all A ∈ A, D ∈ L1 (Ω; B). A

A

E[D|A] is called conditional expectation of D. Let H be a real separable Hilbert space as, for example, deﬁned in (5.4). We call the map W : Ω → H an H-valued random variable if W is F /B(H)measurable. All concepts for Banach-space-valued random variables carry over naturally to Hilbert-space-valued random variables. With the help of Bochnerintegrable random variables we will now introduce the class of mixingale processes, a notion ﬁrst introduced by McLeish (1975) and later extended to Hilbert-space valued processes by Chen & White (1996). Deﬁnition 5.2. Let {Wt }t∈N be a sequence of H-valued random variables on (Ω, F , P) and {At }t∈Z be a ﬁltration of non-decreasing sub-σ-algebras of F . Then {Wt , At } is called an Lp (Ω; H) mixingale (process) if there exists a bounded sequence of non-negative real constants {dt }t∈N and a sequence of non-negative real constants {δk }k∈N with δk → 0 as k → ∞, such that for each t, k ≥ 0, ' ' ' ' 'E[Wt |At−k ]' ≤ dt δk and 'Wt − E[Wt |At+k ]' ≤ dt δk+1 . Lp Lp !∞ If, in addition, k=0 δkς < ∞ for some ς with 0 < a < (1/ς), then {Wt , At } is called a mixingale of size −a.

5.3 Some Probabilistic Prerequisites

101

If Wt is At measurable, then {Wt , At } is an adapted Lp (Ω; H) mixingale and the condition on Wt − E[Wt |At+k ] Lp holds automatically. It suﬃces to have δk = o(k −1/ς ) for some a < (1/ς) or δk = O(k −λ ) for some λ > a to obtain a mixingale {Wt , At } of size −a. The sequence {dt }t∈N is a sequence of magnitude indices and {δk }k∈N measures the decay of E[Wt |At−k ] Lp to zero as k → ∞. This stipulates a form of asymptotic martingale diﬀerence property. As special cases, mixingale processes include independent processes, martingale diﬀerence sequences, and certain types of moving average processes, cf. Gallant & White (1988) Exploiting certain trade-oﬀs between the sequences {dt }t∈N and {δk }k∈N , one obtains various strong and weak laws of large numbers which are found in Chen & White (1996). We need their Corollary 3.8 (ii) (see p. 291) which is formulated next. of Deﬁnition Lemma 5.2. Let {Wt , At } be a Lp (Ω; H) mixingale !in∞the sense 2 5.2 with p ≥ 2 and δk = O k −1/2 (log k)−2 . If t=0 (at dt ) < ∞, where {at }t∈N is a decreasing sequence of positive real numbers with at → 0 as t → ∞, then t ' ' ' ' Wk ' → 0 P − a.s. as t → ∞. 'at k=0

H

To introduce a certain class of mixingales, we need a notion of stochastic dependence. Let A and C be two sub-σ-algebras of F . As two measures of dependence deﬁne # " α(A, C) := sup |P(A ∩ C) − P(A)P(C)| A ∈ A, C ∈ C , " # φ(A, C) := sup |P(C|A) − P(C)| A ∈ A, P(A) > 0, C ∈ C . With these two measures of dependence, we introduce the notion of a mixing sequence of random variables. Deﬁnition 5.3. Let (Ω, F , P) be a probability space, B be a real separable Banach space, and {Dt }t∈Z be a sequence of B-valued random variables deﬁned on (Ω, F , P). Denote Aba := σ(Dt | a ≤ t ≤ b) for all a ≤ b, a, b ∈ Z and deﬁne αk := sup{α(At−∞ , A∞ t+k )} t∈Z

and

φk := sup{φ(At−∞ , A∞ t+k )}. t∈Z

(i) If limk→∞ αk = 0, then {Dt }t∈Z is called a strongly or α-mixing sequence of B-valued random variables. (ii) If limk→∞ φk = 0, then {Dt }t∈Z is called a uniformly or φ-mixing sequence of B-valued random variables. The mixing property of sequences deﬁned in Deﬁnition 5.3 is a form of asymptotic independence. Mixing sequences of random variables constitute

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5 Nonparametric Adaptive Learning

another class of mixingales. Finally, we introduce the notion of near-epoch dependency for Hilbert-space valued random variables. The following deﬁnition is taken from Chen & White (1996). Deﬁnition 5.4. Let (Ω, F , P) be a probability space, B be a real separable Banach space, and H be a real separable Hilbert space. Let {Dt }t∈Z be a sequence of B-valued random variables and {Wt }t∈N be a sequence of H-valued random variables deﬁned on (Ω, F , P), where At+k t−k := σ(Ds |t − k ≤ s ≤ t + k) as before. Then {Wt }t∈N is called Lp (Ω; H)-near-epoch dependent (NED) on {Dt }t∈Z if the following conditions hold: (i) Wt Lp < ∞ for each t ∈ N. (ii) There exists a sequence of non-negative numbers {δk }k∈N with δk → 0 as k → ∞ and a sequence of non-negative numbers {dt }t∈N such that ' ' 'Wt − E[Wt |At+k ]' ≤ dt δk for all t, k ≥ 0. t−k L p

!∞

If, in addition, k=0 δkς < ∞ for some ς with 0 < a < (1/ς), then {Wt }t∈N is called Lp (Ω; H)-NED on {Dt }t∈Z of size −a. It again suﬃces to have δk = o(k −1/ς ) for some a < (1/ς) or δk = O(k −λ ) for some λ > a in order for last condition in Deﬁnition 5.4 to be satisﬁed. The sequence {dt }t∈N is a sequence of magnitude indices and {δk }k∈N measures the intertemporal dependence of {Wt }t∈N on {Dt }t∈Z . Near-epoch dependent processes form another class of mixingales. In particular, we have the following lemma which is given as Lemma 4.3 in Chen & White (1996, p. 293). Lemma 5.3. Let {Dt }t∈Z be a sequence of B-valued random variables on (Ω, F , P) which satisﬁes one of the following two conditions: (i) Either {Dt }t∈Z is strongly mixing with αk of size −b p/( − p) with > p, or (ii) {Dt }t∈Z is uniformly mixing with φk of size −b /( − 1), where > 1. Suppose {Wt }t∈N is a sequence of H-valued random variables in L (Ω; H) with zero means which is Lp (Ω; H)-NED on {Dt }t∈Z of size −a, where ≥ p ≥ 1. Then {Wt , At }, where At := σ(Dk | − ∞ < k ≤ t), is an Lp (Ω; H) mixingale of size − min{a, b}.

5.4 Suﬃcient Conditions for Contractions In this section we formulate a set of more tractable conditions which assure almost-sure convergence of our learning scheme (5.9). These are a specialization of those given in Section 5.2 and are easier to verify and interpret. They are designed to assure convergence of the estimations generated by (5.9) in the case in which the economic law is a contraction. The key idea is that the

5.4 Suﬃcient Conditions for Contractions

103

forecasts as inputs of the economic law are censored in such a way that the stochastic process generated by (5.9) becomes near-epoch dependent. Let S be the space of feasible forecasting rules introduced in Section 5.2. For simplicity of the exposition, we assume throughout this section that censoring is carried out by the time invariant censor map (5.14) as introduced in Example 5.1, so that Υt ≡ Υ . The following conditions are a modiﬁcation of conditions given in Chen & White (1998, Sec. 3) and Chen & White (2002, Sec. 2). These are listed with some redundancy to enhance readability. All proofs of this Section are given in Section 5.5.3 below. Assumption 5.10. The exogenous noise is driven by a sequence of random variables {ξt }t∈N on a complete probability space (Ω, F , P) with values in a subset Ξ ⊂ Rdξ , such that for each t ∈ N, ξt : Ω → Ξ. Moreover: (i) {Dt }t∈Z is a sequence of B-valued random variables on (Ω, F , P) which is either strongly mixing with αk of size −1 or uniformly mixing with φk of size − 12 , and (ii) the process {ξt }t∈N is L2 (Ω; Rdξ )-NED on {Dt }t∈Z of size − 21 and for each t ∈ N, ξt ∈ L (Ω; Rdξ ) for some ≥ 2. Assumption 5.11. The economic law G as given in (2.8) satisﬁes the following conditions. (i) X × Σ ⊂ RdH is open, convex, and has the cone property. (ii) For each (x, y e ) ∈ X × Ym , the map G(·, x, y e ) : Ξ → X satisﬁes the Lipschitz condition ˜ x, y e ) ≤ c1 ξ − ξ , ˜

G(ξ, x, y e ) − G(ξ,

ξ, ξ˜ ∈ Ξ,

uniformly in (x, y e ) with Lipschitz constant c1 ≥ 0. (iii) For each ξ ∈ Ξ, the map G(ξ, ·) : X × Ym → X is Lipschitz continuous uniformly in ξ ∈ Ξ, that is, there exists constants 0 ≤ c2 < 1 and c3 ≥ 0, independent of ξ, such that for each (x, y e ), (˜ x, y˜e ) ∈ X × Ym , ˜ + c3 y e − y˜e .

G(ξ, x, y e ) − G(ξ, x˜, y˜e ) ≤ c2 x − x The important part of Assumption 5.11 is that the economic law is assumed to be contractive with respect to the endogenous variable x. The following three assumptions are a mere repetition of Assumptions 5.3, 5.5, and 5.6. Assumption 5.12. There exists a measurable map M : H → H such that the following holds: (i) M has a unique zero ψ ∈ H, that is, M (ψ ) = 0. Ψ = I(ψ ) is the unique unbiased no-updating forecasting rule associated with the economic law G.

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5 Nonparametric Adaptive Learning

(ii) M is uniformly continuous on bounded sets. That is, for each bounded subset A ⊂ H and each > 0, there exists δ = δ(, A) such that ˜ H < δ implies M (ψ) − M (ψ)

˜ H<

ψ − ψ

for all ψ, ψ˜ ∈ A. Assumption 5.13. There exists a bounded and twice continuously Fr´echet diﬀerentiable functional V : H → R, also referred to as Liapunov function. Let V denote the gradient of the Fr´echet derivative of V. Suppose that V satisﬁes the following properties: (i) V(ψ ) = 0 and

lim

ψH →∞

V(ψ) = ∞.

(ii) V(ψ) > 0 and V (ψ), M (ψ)H < 0 for all ψ ∈ H, ψ = ψ . (iii) V(ψ) < dV := inf{V(ψ) | ψ H > ¯b}. Assumption 5.14. {at }t∈N is a sequence of exogenously given non-increasing positive numbers such that the following conditions hold: !∞ (i) at → 0 as t → ∞, (ii) t=0 at = ∞, (iii) at ≤ at+1 (1 + a at )

for some 0 < a ≤ 1.

The assumptions on the updating functions are modiﬁed as follows. Assumption 5.15. The updating functions {Mt }t∈N given in Assumption 5.4 satisfy the following conditions. (i) The sequence {Mt }t∈N is bounded on bounded sets uniformly in t. That is, for any bounded subset B ⊂ X × Σ × H, there exists a constant cB such that for each t ∈ N,

Mt (x, Z, ψ) H ≤ cB

for all (x, Z, ψ) ∈ B.

(ii) For any bounded subset D ⊂ X × Σ × H, there exists a sequence of positive real numbers {c4,t }t∈N such that for each t ∈ N and arbitrary (x, Z, ψ), ˜ ∈ D, ˜ ψ) (˜ x, Z, ' '

' ˜ H . ˜ H ≤ c4,t '(x, Z) − (˜ 'Mt (x, Z, ψ) − Mt (˜ ˜ ' + ψ − ψ

˜ ψ)

x, Z) x, Z, The sequence {c4,t }t∈N satisﬁes ∞

(c4,t at )2 < ∞

t=0

with {at }t∈N as given in Assumption 5.14.

5.4 Suﬃcient Conditions for Contractions

105

(iii) For each (x0 , Z0 ) ∈ D, any ω ∈ Ω, and any diﬀerentiable ψ ∈ H with

ψ H ≤ bH , where bH ≥ bγmax with bγmax as given by Proposition 5.1, ' '

lim 'E Mt G t + 1, ·, x0 , Z0 , Υ (ψ) , ψ − M (ψ)'H = 0. t→∞

It is straightforward to verify that Conditions (i) and (ii) of Assumption 5.15 imply Conditions (i)-(iii) of Assumption 5.8. The main task is to show that Assumptions 5.10-5.15 imply Assumption 5.8 (iv) and Assumption 5.9. We begin with the latter Assumption which is easier to establish. The following Lemma 5.4 demonstrates that the long-run deviating eﬀect on realizations for two diﬀerent sequences of forecasting rules can be controlled under the hypotheses of Assumption 5.11. Lemma 5.4. Let the hypotheses of Assumption 5.11 be satisﬁed and {Ψi }i∈N and {Ψ˜i }i∈N be two sequences of forecasting rules in S. For for each t ∈ N, x0 , Z˜0 ) ∈ X × Σ, and each ω ∈ Ω, set arbitrary initial conditions (x0 , Z0 ), (˜ (xt , Zt ) = G(t, ω, x0 , Z0 , Ψ0t−1 )

and

Then

xt − x ˜t ≤ ct2 x0 − x˜0 + c3

(˜ xt , Z˜t ) = G(t, ω, x ˜0 , Z˜0 , Ψ˜0t−1 ). t−1

ci2 Ψt−i − Ψ˜t−i ∞ .

i=0

Lemma 5.4 implies that Assumption 5.2 (iii) can be satisﬁed by ensuring that all forecasts remain bounded. It follows that a bounded subset D ⊂ X×Σ which is forward invariant under G ·, Ψ for all forecasting rules Ψ ∈ S which are uniformly bounded exists. Using the Sobolev Imbedding Theorem stated in Section 5.5.1 below, Lemma 5.4 is now applied to establish Assumption 5.9. Proposition 5.2. If Assumption 5.11 is satisﬁed, then Assumption 5.9 is satisﬁed as well. As regards Assumption 5.8 (iv), we start with the observation that this assumption is satisﬁed if the endogenous variables together with the forecasts form a near-epoch dependent process. Proposition 5.3. Let Assumptions 5.10–5.15 be satisﬁed. If, in addition, for each diﬀerentiable ψ ∈ H with ψ H ≤ bH , the sequence of functions ,

G t, ·, x0 , Z0 , Υ (ψ) t∈N is L2 (Ω; RdH )-NED on {Dt }t∈Z of size − 21 in the sense of Deﬁnition 5.4, then Assumption 5.8 (iv) is satisﬁed.

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5 Nonparametric Adaptive Learning

With this result, the key idea to ensure that Assumption 5.8 (iv) holds is to censor the forecasts in such a way that the process of endogenous variables and forecasts {(xt , yte )}t∈N inherits the near-epoch dependency of the exogenous noise process imposed by Assumption 5.10. One way of doing so is to apply Proposition 4.4 in Kuan & White (1994) stating that contractions inherit the near-epoch dependency of the exogenous perturbations. This result will be exploited in the proof of the following proposition. Proposition 5.4. Let the hypotheses of Assumptions 5.10 and 5.11 be satisﬁed and (x0 , Z0 ) ∈ X × Σ and ω ∈ Ω be arbitrary initial conditions. Suppose Ψ ∈ S is a forecasting rule such that the sequence {yte }t∈N is L2 (Ω; Rdy )-NED on {Dt }t∈Z of size − 21 in the sense of Deﬁnition 5.4, where

xt = G ξt (ω), xt−1 , Ψ (xt−1 , Zt−1 ) . e = Ψ (xt−1 , Zt−1 ) yt−1 Then the sequence of functions ,

G t, ·, x0 , Z0 , Ψ ) t∈N is L2 (Ω; RdH )-NED on {Dt }t∈Z of size − 21 . In order to manipulate the least amount of forecasts necessary, we set r = 0 and modify the censor map (5.14) as introduced in Example 5.1 as follows. Recall that by Remark 5.2 it suﬃces to consider the space of bounded p (X × Ym−1 ; Rdy ). Choose continuously diﬀerentiable forecasting rules ψ ∈ CB a positive constant 0 < cΥ < 1 and redeﬁne (5.13) as 1 1 (X × Ym−1 ; Rdy ) −→ CB (X × Ym−1 ; Rdy ) CB Υ : (5.17) ψ −→ Υ (ψ) by setting

Υ (ψ) =

ψ cΥ Dψ∞

ψ

if Dψ ∞ < cΥ . if Dψ ∞ ≥ cΥ

As before, (5.17) maps diﬀerentiable functions onto diﬀerentiable contractions with prescribed Lipschitz constant cΥ . For each t ∈ N, deﬁne p (X × Ym−1 ; Rdy ) → S, Υ : CB

ψ → I ◦ Υ (ψ),

(5.18)

where I is deﬁned in (5.5). By Lemma 5.6 of Section 5.5.1 below, the censor map (5.18) satisﬁes Assumption 5.2 (ii). The following proposition shows that with the help of the censor map (5.18), the hypotheses of Proposition 5.4 can be fulﬁlled.

5.4 Suﬃcient Conditions for Contractions

107

Lemma 5.5. Let the hypotheses of Assumptions 5.10 and 5.11 be satisﬁed. Suppose, in addition, that c2 = c3 < 1 and let r = 0, so that Σ = Ym and r = 0. Choose the Lipschitz constant of the censor map Υ deﬁned in (5.18) p (X × Ym−1 ; Rdy ) such that (m − 1)cΥ + c2 < 1. Then each Ψ = Υ (ψ), ψ ∈ CB satisﬁes the hypothesis of Proposition 5.4, so that the conclusion of Proposition 5.4 holds. Observe that the Lipschitz constant cΥ cannot be chosen independently of the economic law G so that a-priori information on the Lipschitz constant c2 of G is necessary to apply Lemma 5.5. On the other hand, it is straightforward to generalize Lemma 5.5 to the case r > 0. The almost-sure convergence of the learning scheme (5.9) under the more specialized assumptions of this section is now obvious as all crucial assumptions of Section 5.2 follow from Propositions 5.2 and 5.3 and Lemma 5.5 together with Theorem 5.2. Theorem 5.3. Let Assumptions 5.10-5.13 be fulﬁlled and assume, in addition, that all hypotheses of Lemma 5.5 are satisﬁed. Then Assumptions 5.15.5 and Assumption 5.7 are fulﬁlled and all conclusions of Theorem 5.2 hold. In particular, if (x0 , Z0 ) ∈ D and ψˆ0 ∈ H0 are arbitrary initial conditions, then the sequence {ψˆt }t∈N of functions in H generated by the learning scheme (5.9) converges, i.e., lim ψˆt − ψ H = 0

t→∞

P − a.s.

If, in addition, Υ (ψ ) = Ψ , then lim Ψt − Ψ H = 0

t→∞

P − a.s.

Under the hypotheses of Theorem 5.3, an approximated unbiased forecasting rule is only applied, if the system under unbiased expectations is suﬃciently contractive. In this case the contraction property of Theorem 4.3 in Section 4.4 of Chapter 4, guaranteeing the existence of a random ﬁxed point, is satisﬁed. In other words, the hypotheses of Theorem 5.3 are suﬃcient to estimate and thus learn an unbiased forecasting rule of an economic random dynamical system which admits a random ﬁxed point under rational expectations.

Concluding Remarks Based on nonparametric estimation techniques, this chapter developed a learning scheme for nonlinear economic systems with expectations feedback. Combining the notion of an economic law with the nonparametric estimation techniques, we established conditions under which strongly consistent estimates of

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5 Nonparametric Adaptive Learning

an unbiased no-updating rule exist. A key element of the method was to censor the forecasts in such a way that the resulting process has properties which ensure consistent estimators. Our approach opens up the possibility to select an approximation of a preferred forecasting rule on economic grounds, as soon as parameter estimates are suﬃciently precise. It thus provides the missing link for an appropriate treatment of multiple solutions which in models with expectational leads will occur generically. An open issue is the case in which the dynamics under unbiased expectations is not governed by a contraction. This requires a method which allows to remove the restriction of the Lipschitz constant of the applied forecasting rule as soon as estimators are suﬃciently precise in the sense of an ε-unbiased forecasting rule introduced in Chapter 4. As in the linear case, this requires knowledge on the asymptotic behavior of the estimation procedure. Asymptotic properties such as convergence rates of Robbins-Monro procedures are known (e.g., see Chen & White (2002) and Yin & Zhu 1990), the application to the present setup, however, is left for future research. Summarizing, this chapter conﬁrms the lesson from the linear case. A careful distinction between four separate issues is important for plausible learning schemes for economic systems with expectations feedback. First, the existence of a desired forecasting rule; second, the dynamic stability of the system under this forecasting rule; third, the dynamic stability of the system under the applied learning scheme and, ﬁnally, the success of the learning scheme in terms of strongly consistent estimates. In essence, the mathematical object to be estimated from time series data is the economic law.

5.5 Mathematical Appendix 5.5.1 Sobolev Imbedding Theorem We begin with a brief review of all necessary properties of Sobolev spaces. All details for the case of R-valued functions are found in Adams (1975). The extension to vector-valued functions required for this section is straightforward, see also Yosida (1978). In view of the Sobolev imbedding theorem, which will be stated below we require that, in addition, X × Σ satisﬁes the cone property. Recall to this end that given a point ζ ∈ RdH , an open ball B1 with center x, and an open ball B2 not containing ζ, the set " # Cζ := B1 ∩ ζ + λ(η − ζ) ∈ RdH η ∈ B2 , λ > 0 is called a ﬁnite cone in RdH with vertex ζ. The set X×Σ has the cone property if there exists a ﬁnite cone C such that each point ζ ∈ X × Σ is the vertex of a ﬁnite cone Cζ ⊂ X × Σ which is congruent to C. For sake of completeness we deﬁne the segment property of a set. This property is required, for instance, to show that the space of diﬀerentiable

5.5 Mathematical Appendix

109

functions with compact support is a dense subspace of a Sobolev space (e.g., Thm. 3.18, p. 54 Adams 1975). Denote by X × Σ the closure of X × Σ and by ∂(X × Σ) its boundary. X × Σ has the segment property if there exists a locally ﬁnite open cover {Uj } of ∂(X × Σ) and a corresponding sequence {ηj } nonzero vectors such that if ζ ∈ X × Σ ∩ Uj for some j, then ζ + tηj ∈ X × Σ for all 0 < t < 1. Loosely speaking, the boundary of X × Σ must be (dH − 1)dimensional and cannot simultaneously lie on both sides of any given part of its boundary. If, in addition, X × Σ is bounded, then the cone and the segment condition are implied by the simple condition that ∂(X × Σ) is a locally Lipschitz boundary, that is, each point on the boundary has a neighborhood which can be represented by the graph of a Lipschitz continuous function. Further details may be found in Adams (1975, Chap. IV, pp. 65-67). An Rn -valued function f : X → Rn is called square integrable if $0 % 21

f 2 :=

f (ζ) 2 dζ , X×Σ

where dζ means integration with respect to the Lebesgue measure. Denote by α = (α1 , . . . , αdH ) a dH -tuple of nonnegative integers αj , also referred to as !d H multi-index, and set |α| = j=1 αj . A diﬀerential operator Dα of order |α| is then deﬁned by setting Dα f (ζ) :=

∂ |α| αd f (ζ). ∂ζ1α1 · · · ∂ζdH H

Put D(0,...,0) f (ζ) = f (ζ) and denote the space of all p-times continuously diﬀerentiable Rn -valued functions by C p (X × Σ; Rn ). Deﬁne a norm on this space by % 21 $

f p :=

Dα f 22 . (5.19) 0≤|α|≤p

The completion of the set " # f ∈ C p (X × Σ; Rn ) f p < ∞ with respect to the norm (5.19) is commonly referred to as the Sobolev space Wp,2 (X × Σ; Rn ). By Theorem 3.5 on p. 47 of Adams (1975), each space Wp,2 (X × Σ; Rn ), 1 ≤ p < ∞, is a separable Hilbert space. For any j ≥ 0, consider the norm

f ∞,j := max

sup Dα f (ζ)

0≤α≤j ζ∈X×Σ

(5.20)

for some j-times continuously diﬀerentiable function f . For notational convenience, put · ∞ ≡ · ∞,0 . Then the space

110

5 Nonparametric Adaptive Learning

" # j CB (X × Σ; Rn ) := f ∈ C j (X × Σ; Rn ) f ∞,j < ∞ is a Banach space when endowed with the norm · ∞,j . j (X × Σ; Rn ) is meant that By an imbedding of Wj+p,2 (X × Σ; Rn ) into CB j+p,2 n (X × Σ; R ) when interpreted as a function can be redeﬁned each f ∈ W on a set of measure zero in X × Σ in such a way that the modiﬁed function f˜ j [which equals f in Wj+p,2 (X×Σ; Rn )] belongs to CB (X×Σ; Rn ) with an upper bound for its norm. This result is known as the Sobolev Imbedding Theorem. The version needed in these notes is stated next. Theorem 5.4. Let X × Σ be an open subset of RdH which has the cone property. Suppose 2p > dH . Then the imbedding Wj+p,2 (X × Σ; Rn ) → j (X × Σ; Rn ) is continuous in the sense that CB

f ∞,j ≤ K f j+p

(5.21)

for all f ∈ Wj+p,2 (X × Σ; Rn ), j ≥ 0. The constant K depends only on X × Σ and the parameters p and j. The proof of Theorem 5.4 and further details are found in Adams (1975, Chap. V, p. 95). With this brief review all proofs of Section 5.4 can be carried out, setting H = Wp,2 (X × Σ; Rdy ) and · H = · p . We begin with a fundamental property of the censor map (5.14) Lemma 5.6. Let Assumption 5.2 (i) be satisﬁed and let A ⊂ H be a subset which is bounded with respect to the norm · H . Then the following holds. (i) The censor map as deﬁned in (5.14) can be continuously extended to a map Υ : H → H which is Lipschitz continuous when restricted to A, so that ˜ H for all ψ, ψ˜ ∈ A ˜ H ≤ K H ψ − ψ

Υ (ψ) − Υ (ψ)

Υ with a suitable Lipschitz constant KΥH . (ii) There exists a constant KΥ such that Υ when restricted to A is Lipschitz continuous with respect to the supremum norm, i.e., ˜ H ˜ ∞ ≤ KΥ ψ − ψ

Υ (ψ) − Υ (ψ)

for all ψ, ψ˜ ∈ A.

p Proof. Deﬁne a map F : CB (X × Σ; Rdy ) → R+ by 1 if Dψ ∞ < cΥ , F (ψ) = cΥ if Dψ ∞ ≥ cΥ . Dψ∞

with the constant 0 < cΥ < 1 as chosen above. Then the censor map (5.14) takes the form Υ (ψ) = I F (ψ)ψ . We show that Υ is Lipschitz continuous

5.5 Mathematical Appendix

111

with respect to the norms · H and · H when restricted to · H -bounded p subsets of CB (X × Σ; Rdy ). Since 0 < F (ψ) ≤ 1 and I(ψ) H ≤ ψ H , we have ˜ H + |F (ψ) − F (ψ)| ˜ ψ

˜ H. ˜ H ≤ ψ − ψ

(5.22)

Υ (ψ) − Υ (ψ)

There are essentially three cases. Υ is clearly Lipschitz continuous in the ﬁrst ˜ ∞ < cΥ . In the second one Dψ ∞ , Dψ

˜ ∞ ≥ cΥ one when Dψ ∞ , Dψ

˜ and in the third one Dψ ∞ < cΥ ≤ Dψ ∞ . In the second case, Dψ

˜ ∞ − Dψ ∞ ˜ ≤ cΥ ˜ ∞. |F (ψ) − F (ψ)| (5.23) ≤ 1 Dψ − Dψ

˜ ∞ Dψ ∞ cΥ Dψ

In the third case, |F (ψ) − 1| ≤

˜ ∞

Dψ ∞ − Dψ

≤

Dψ ∞

1 cΥ

˜ ∞.

Dψ − Dψ

(5.24)

Since 2p > dH , we may apply Theorem 5.4 for j = 0 to (5.23) and (5.24) to obtain ˜ ≤ 1 Dψ − Dψ

˜ ∞ ≤ K ψ − ψ

˜ H |F (ψ) − F (ψ)| Υ cΥ

for some suitable constant KΥ . Inserting into (5.22) gives ˜ H ≤ 1 + K ψ

˜ H ψ − ψ

˜ H.

Υ (ψ) − Υ (ψ)

Υ

This shows that Υ is Lipschitz continuous on · H -bounded subsets. Since p CB (X × Σ; Rdy ) is a dense subset of H, Υ can be continuously extended to H. This proves the ﬁrst assertion. The second assertion then follows again from Theorem 5.4.

5.5.2 Proofs of Section 5.2 We will preface the proof of Proposition 5.1 by an inﬁnite-dimensional version of Lemma D on p. 147 of Metivier & Priouret (1984). Its proof follows along the same lines and is omitted. Lemma 5.7. Let {at }t∈N be a sequence of numbers which satisﬁes Assumption 5.3. Let A be some ‘index set’ and for each Φ ∈ A, {Lt (Φ)}t∈N be a sequence of H-valued random variables on the probability space (Ω, F , P), such that t ' ' ' Lj (Φ)'H = 0 P − a.s. lim sup at t→∞ Φ∈A

j=1

Then for each α > 0, there exists a random variable N (α, ω), such that for all m > t > N (α, ω), m−1 $ % ' 'm−1 sup ' aj aj Lj (Φ)'H ≤ α 1 +

Φ∈A

j=t

j=t

P − a.s.

112

5 Nonparametric Adaptive Learning

Proof of Proposition 5.1. Step 1. Assume on the contrary that γmax = ∞, so that the sequence of estimates {ψˆt }t∈N crosses the sphere {ψ ∈ H| ψ H ≤ ¯b} inﬁnitely often. As a consequence {ψˆt }t∈N cannot converge to ψ so that by Assumption 5.6 (ii), there exists η > 0 such that |V (ψˆt ), M (ψˆt )H | > η

P − a.s. for all t ∈ N.

(5.25)

By Assumption 5.6 (iii), (V(ψ), dV ) is a non-empty open interval. It follows from Assumption 5.2 (iii) and Assumption 5.8 (i) that there exists cM such that ' ' 'Mt (x, Z, ψ)' ≤ cM for all (x, Z) ∈ D, ψ ≤ ¯b. (5.26) H For δ > 0, choose t0 large enough so that at < δ for all t ≥ t0 and σ1 ≤ σ2 with [σ1 , σ2 ] ⊂ (V (ψ), dV ). Deﬁne # " t1 := min t ≥ t0 V(ψˆt ) ≥ σ1 , # " t2 := min t ≥ t0 V(ψˆt ) ≥ σ2 . By our initial assumption, t1 and t2 are well deﬁned and ﬁnite. t1 describes the ﬁrst entry of the sequence V(ψˆt ) into the interval [σ1 , σ2 ] after time t0 t2 −1 and t2 the ﬁrst exit. The sequence {ψˆt }t=t is bounded by ¯b. (If this were 0 not the case and ψˆt H > ¯b for some t1 ≤ t < τ2 , then V(ψˆt ) > dV , a contradiction.) The continuity of V and (5.26) imply that δ and hence t0 can be chosen such that t2 − t1 > 1. For τ > δ and i ≥ t0 , set n−1 " # n(i, τ ) := max n > i aj ≤ τ .

(5.27)

j=i

Then for τ > δ small enough, we have t1 − 1 < n(t1 − 1, τ ) < t2 . As a consequence, V(ψˆn(t1 −1,τ ) ) ∈ [σ1 , σ2 ]. We next will show in two steps that V(ψˆn(t1 −1,τ ) ) < σ1 . This contradicts the assumption γmax = ∞ and thus proves the assertions of the proposition. Step 2. We will next apply Lemma 5.7 several times. From Step 1. we now that τ > δ are such that no truncations occur between t = i ≥ t1 − 1 and t = n(i, τ ). Let n ≤ n(i, τ ). In this case ψˆn − ψˆi =

n−1

aj Mj (G(j + 1, ω, x0 , Z0 , Ψ0j ), ψˆj ).

j=i

Since {ψˆt }nt=i is bounded by ¯b, it follows from (5.26) that

(5.28)

5.5 Mathematical Appendix

ψˆn − ψˆi H ≤

n−1

aj cM ≤ cM τ.

113

(5.29)

j=i

For each j > i, write j Ψ0i := (Ψ0 , . . . , Ψi , Ψi , . . . , Ψi )

(5.30)

(j−i)−times

for a series of forecasting rules which is constant after the i-th entry. Decompose the sum of the r.h.s. of (5.28) into ψˆn − ψˆi = So (i, n) +

n−1

aj M (ψˆi ),

(5.31)

j=i

where So (i, n) = S1 (i, n) + S2 (i, n) + S3 (i, n) with S1 (i, n) :=

n−1

aj Mj (G(j + 1, ω, x0 , Z0 , Ψ0j ), ψˆj )

j=i

S2 (i, n) :=

n−1

j −Mj (G(j + 1, ω, x0 , Z0 , Ψ0i ), ψˆj ) ,

j aj Mj (G(j + 1, ω, x0 , Z0 , Ψ0i ), ψˆj )

j=i

S3 (i, n) :=

n−1

(5.32)

(5.33)

j −Mj (G(j + 1, ω, x0 , Z0 , Ψ0i ), ψˆi ) ,

j aj Mj (G(j + 1, ω, x0 , Z0 , Ψ0i ), ψˆi ) − M (ψˆi ) .

(5.34)

j=i

The ﬁrst term (5.32) takes account of the expectations feedback. By Assumption 5.9, for any > 0, there exists δ = δ () such that j

G(j + 1, ω, x0 , Z0 , Ψ0j ) − G(j + 1, ω, x0 , Z0 , Ψ0i ) ≤ .

for all j = i, . . . , n − 1, provided that τ ≤ cδM . Choose α > 0. By Assumption 5.8 (ii) and Lemma 5.7, there exists N1 (α, ω) > t0 such that for each i > N1 (α, ω) and each n ≤ n(i, τ ),

S1 (i, n) H ≤

n−1

aj mj (ψˆj ) ≤ m τ

P − a.s.

j=i

Concerning the second term (5.33), write for simplicity of notation

j xj+1 (ω), Zj+1 (ω) = G(j + 1, ω, x0 , Z0 , Ψ0i ) for j ≥ i. Assumption 5.8 (iii) and (5.29) then imply P-a.s.

(5.35)

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5 Nonparametric Adaptive Learning

S2 (i, n) H ≤

n−1

aj hj+1 xj+1 (ω), Zj+1 (ω) ψˆj − ψˆi H

j=i

≤

n−1

aj hj+1 xj+1 (ω), Zj+1 (ω) − E hj+1 xj+1 (·), Zj+1 (·) ψˆj − ψˆi H

j=i

+

n−1

aj E hj+1 xj+1 (·), Zj+1 (·) ψˆj − ψˆi H .

j=i

By Lemma 5.7, there exists N2 (α, ω) > N1 (α, ω) such that for each i > N2 (α, ω) and each n ≤ n(i, τ ),

S2 (i, n) H ≤ αcM (1 + τ )τ + hcM τ 2

P − a.s.

(5.36)

Concerning the third term (5.34), Assumption 5.8 (iv) and Lemma 5.7 imply the existence of N3 (α, ω) > N2 (α, ω) such that for each i > N3 (α, ω) and each n ≤ n(i, τ ),

S3 (i, n) H ≤ α(1 + τ ) P − a.s. (5.37) Combining the estimates (5.35)-(5.37), we get

So (i, n) H ≤ α(1 + τ + cM τ ) + m τ + (α + h)cM τ 2

P − a.s.

(5.38)

Step 3. The Taylor expansion of V at ψˆt1 −1 gives V(ψˆn(t1 −1,τ ) ) =V(ψˆt1 −1 ) + V (ψˆt1 −1 ), [ψˆn(t1 −1,τ ) − ψˆt1 −1 ]H + R2 (ψˆt −1 , ψˆn(t −1,τ ) − ψˆt −1 ), 1

where

0 R2 (ψ, ϕ) =

1

(5.39)

1

1

(1 − s)D2 (ψ + sϕ)(ϕ, ϕ)ds,

ψ, ϕ ∈ H,

0

e.g., see Lang (1968, Chap. XVI) or Berger (1977).4 In view of (5.29), there exists κ2 so that 0 1 (1 − s)ds |R2 (ψˆt1 −1 , ψˆn(t1 −1,τ ) − ψˆt1 −1 )| ≤ κ2 ψˆn(t1 −1,τ ) − ψˆt1 −1 2H 0

≤

κ2 c2M τ 2

(5.40)

Using (5.31), we can decompose the second term in (5.39) into V (ψˆt1 −1 ), [ψˆn(t1 −1,τ ) − ψˆt1 −1 ]H = I +

n(t1 −1,τ )−1

aj V (ψˆt−1 ), M (ψˆt1 −1 )H

j=t1 −1 4

Here D2 V(ψ)(ϕ1 , ϕ2 ) denotes the second Fr´echet derivative at ψ ∈ H applied to (ϕ1 , ϕ2 ) ∈ H × H.

5.5 Mathematical Appendix

with

115

I := V (ψˆt1 −1 ), So (t1 − 1, n(t1 − 1, τ )H .

Using (5.38), we have

|I| ≤ κ1 α(1 + τ + cM τ ) + mτ + (α + h)cM τ 2 with

(5.41)

, κ1 := sup V (ψ) H ψ H ≤ ¯b .

Observe that by deﬁnition of n(t1 − 1, τ ) in (5.27), n(t1 −1,τ )−1

τ − an(t1 −1,τ )−1 ≤

aj ≤ τ.

j=t1 −1

Hence, inserting (5.40) and (5.41) into (5.39), we get V(ψˆn(t1 −1,τ ) ) ≤ σ1 + |I| − η[τ − an(t1 −1,τ )−1 ] + κ2 c2M τ 2 ≤ σ1 + ακ1 [1 + τ + cM τ ] + η an(t1 −1,τ )−1 + (κ2 c2M + (α + h)κ1 cM )τ − η + mκ1 ) τ for η as deﬁned in (5.25). Let <

η mκ1

" and τ < min

η−mκ1 (κ2 cM +(α+h)κ1 )cM

, cδM

#

and choose α and δ such that ακ1 [1 + τ + cM τ ] + δη < η − mκ1 − (κ2 cM + (α + h)κ1 )cM τ τ.

(5.42)

By our initial assumption we may choose t0 so large that at < δ for all t ≥ t0 . Hence (5.42) implies V(ψˆn(t1 −1,τ ) ) < σ1 which proves our contradiction stated at the end of Step 1.

Proof of Theorem 5.2. The proof of the proposition relies on a repeated application of Lemma 5.7. Step 1. Recall that by Assumption 5.2 (iii), the sequence # " (5.43) G(t + 1, ω, x0 , Z0 , Ψ0t ) t∈N

ˆ t ∈ N remains bounded in some set D. By Proposition with Ψt = Υt (ψ), 5.1 there exists a random integer N0 (ω) from where on no truncation of the learning algorithm occurs such that

ψˆt+1 = ψˆt + at Mt G(t + 1, ω, x0 , Z0 , Ψ0t ), ψˆt for all t ≥ N0 (ω) (5.44)

116

5 Nonparametric Adaptive Learning

and a constant bH so that ψˆt H ≤ bH for all t ≥ N0 (ω). The two boundedness properties (5.43) and (5.44) together with the boundedness property of Mt stated in Assumption 5.8 (i) imply the existence of a constant cM > 0 such that '

' 'Mt G(t + 1, ω, x0 , Z0 , Ψ0t ), ψˆt ' ≤ cM P − a.s. (5.45) H for all t ≥ N0 (ω). Step 2. The idea is to show that Assumption 5.7 holds, as this ensures almost sure convergence by Theorem 5.1. For arbitrary t < n, set S(t, n) :=

n−1

aj Mj (G(j + 1, ω, x0 , Z0 , Ψ0j ), ψˆj ) − M (ψˆj ) .

(5.46)

j=t

In order to verify Assumption 5.7, it suﬃces to show that for arbitrary but ﬁxed T > 0, ' ' lim sup 'S(t, n)'H = 0 P − a.s., (5.47) t→∞ t
where for each t ∈ N, i−1 # " aj ≤ T . n(t, T ) := max i > t

(5.48)

j=t

We will apply Lemma 5.7 several times to prove (5.47). By Assumption 5.5, for each > 0, there exists δ = δ() > 0 so that

M (ψˆi ) − M (ψˆj ) H <

whenever ψˆi − ψˆj H < δ.

(5.49)

holds. Fix Moreover, by Assumption 5.9, there √ exists δ = δ () so that (5.16) √ an arbitrary α ∈ (0, 1) such that α T cM < min{δ, δ }. Set τ = α T and deﬁne recursively

i0 = t, i1 = n(i0 , τ ), . . . , ir = n(ir−1 , τ ), . . .

(5.50)

Using the fact that by Step 1 no truncations occur from N0 (ω) on, it follows from (5.45) that for each i0 = t ≥ N0 (ω), each ir , and each i ∈ (ir , ir+1 ],

ψˆi − ψˆir H ≤

i−1

' ' aj 'Mj (G(j + 1, ω, x0 , Z0 , Ψ0j ), ψˆj )'H

j=ir

≤

i−1 j=ir

Step 3. For each j > i, write

aj cM ≤ τ cM < min{δ, δ }

P − a.s.

(5.51)

5.5 Mathematical Appendix j Ψ0i := (Ψ0 , . . . , Ψi , Ψi , . . . , Ψi )

117

(5.52)

(j−i)−times

for a series of forecasting rules which is constant after the i-th entry. Now let i < i be arbitrary and decompose the sum in (5.46) into four terms S(i, i ) := S1 (i, i ) + S2 (i, i ) + S3 (i, i ) + S4 (i, i ) with and

S1 (i, i ) :=

i −1

aj Mj (G(j + 1, ω, x0 , Z0 , Ψ0j ), ψˆj )

j=i

S2 (i, i ) :=

i −1

(5.53)

j −Mj (G(j + 1, ω, x0 , Z0 , Ψ0i ), ψˆj ) ,

j aj Mj (G(j + 1, ω, x0 , Z0 , Ψ0i ), ψˆj )

j=i

(5.54)

j −Mj (G(j + 1, ω, x0 , Z0 , Ψ0i ), ψˆi ) ,

and S3 (i, i ) :=

i −1

j aj Mj (G(j + 1, ω, x0 , Z0 , Ψ0i ), ψˆi ) − M (ψˆi ) ,

(5.55)

aj M (ψˆi ) − M (ψˆj ) .

(5.56)

j=i

S4 (i, i ) :=

i −1

j=i

The ﬁrst term (5.53) takes account of the expectations feedback. Assumption 5.9 together with (5.51) implies for any ir ≤ j < ir+1 , j ) ≤ .

G(j + 1, ω, x0 , Z0 , Ψ0j ) − G(j + 1, ω, x0 , Z0 , Ψ0i r

By Assumption 5.8 (ii) and Lemma 5.7, there exists N1 (α, ω) > N0 (α, ω) such that for each i0 = t > N1 (α, ω), each ir and each i ∈ (ir , ir+1 ],

S1 (ir , i) H ≤

i−1

aj mj (ψˆj ) ≤ mτ

P − a.s.

j=ir

Concerning the second term (5.54), write for simplicity of notation

j xj+1 (ω), Zj+1 (ω) = G(j + 1, ω, x0 , Z0 , Ψ0i ) for j ≥ i. Assumption 5.8 (iii) and (5.51) then imply P-a.s.

(5.57)

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5 Nonparametric Adaptive Learning

S2 (ir , i) H ≤

i−1

aj hj+1 xj+1 (ω), Zj+1 (ω) ψˆj − ψˆir H

j=ir

≤

i−1

aj hj+1 xj+1 (ω), Zj+1 (ω) − E hj+1 xj+1 (·), Zj+1 (·) ψˆj − ψˆir H

j=ir

+

i−1

aj E hj+1 xj+1 (·), Zj+1 (·) ψˆj − ψˆir H .

j=ir

By Lemma 5.7, there exists N2 (α, ω) > N1 (α, ω) such that for each i0 = t > N2 (α, ω), each ir , and each i ∈ (ir , ir+1 ],

S2 (ir , i) H ≤ α(1 + τ )τ cM + τ 2 hcM

P − a.s.

(5.58)

Concerning the third term (5.55), Assumption 5.8 (iv) and Lemma 5.7 imply the existence of N3 (α, ω) > N2 (α, ω) such that for each i0 = t > N3 (α, ω), each ir , and each i ∈ (ir , ir+1 ],

S3 (ir , i) H ≤ α(1 + τ )

P − a.s.

(5.59)

Concerning the fourth term (5.56), (5.51) together with (5.49) imply

S4 (ir , i) H ≤

i−1

aj M (ψˆi ) − M (ψˆj ) H ≤

j=ir

i−1

aj ≤ τ .

(5.60)

j=ir

Step 4. Combining the estimates (5.60)–(5.57), there exists constants β1 and β2 such that for all ir > i0 = t > N3 (α, ω), ' ' √ sup 'S(ir , i)'H ≤ α β1 + αβ2 P − a.s. ir
It follows from the deﬁnition of n(t, T ) that for all ir > i0 = t > N3 (α, ω), % √ $ ' ' sup 'S(t, n)'H ≤ 1 + √2α ( α β1 + αβ2 ) t
= (2 +

√ √ α)β1 + (α + 2 α)β2 .

Since and α can be made arbitrarily small, this proves (5.47) and the conclusion follows from Theorem 5.1.

5.5.3 Proofs of Section 5.4 Proof of Lemma 5.4. Let (x0 , Z0 ), (˜ x0 , Z˜0 ) ∈ X × Σ, ω ∈ Ω be arbitrary ˜ but ﬁxed and {Ψi }i∈N and {Ψi }i∈N be two arbitrary sequences of forecasting rules in S. Let

5.5 Mathematical Appendix

119

yte = Ψt (xt , Zt ) and y˜te = Ψ˜t (˜ xt , Z˜t ). By assumption e e

xt − x˜t ≤ c2 xt−1 − x ˜t−1 + c3 yt−1 − y˜t−1

and by induction on t,

xt − x ˜t ≤

ct2

x0 − x ˜ 0 + c3

t−1

e e ci2 yt−i − y˜t−i

.

i=0

The assertion then follows from the deﬁnition of the maximum norm · ∞ .

Proof of Proposition 5.2. Let δ > 0 be arbitrary and {ψˆt }t∈N be a sequence of forecasting rules so that

ψˆt − ψˆ0 H < δ

for all t ∈ N.

Put Ψt = Υ (ψˆt ) for each t ∈ N and let (xt , Zt ) = G(t, ω, x0 , Z0 , Ψ0t−1 ) and (˜ xt , Z˜t ) = G(t, ω, x0 , Z0 , Ψ0 ), where e e e e Zt = (yt−1 , xt−1 , . . . , yt−r , xt−r ) and Z˜t = (˜ yt−1 ,x ˜t−1 , . . . , y˜t−r , x˜t−r ).

By Lemma 5.4 and Lemma 5.6, t−1 $ %2 ˜t 2 + yte − y˜te 2 ≤ c3 ci2 Ψt−i − Ψ0 ∞ + Ψt − Ψ0 2∞

xt − x i=0 t−1 %2 $ ≤ c3 K Υ ci2 ψˆt−i − ψˆ0 H + KΥ2 ψˆt − ψˆ0 2H

$ ≤ 1+

i=0 c3 1−c2

%2

(KΥ δ)2 .

√ For arbitrary > 0, choose δ := (1 − c2 )/[KΥ (1 − c2 + c3 ) 1 + r]. Then '

G(t, ω, x0 , Z0 , Ψ0t−1 ) − G(t, ω, x0 , Z0 , Ψ0 )' < and Assumption 5.9 is satisﬁed.

Proof of Proposition 5.3. Step 1. Let (x0 , Z0 ) ∈ X × Σ, ω ∈ Ω, and a diﬀerentiable ψ ∈ H with ψ H ≤ bH be arbitrary. For ease of notation, write

xt (ω), Zt (ω) = G t, ω, x0 , Z0 , Υ (ψ) , t ∈ N

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5 Nonparametric Adaptive Learning

and for arbitrary t < n, set S(t, n) :=

n−1

aj Mj (xj+1 (ω), Zj+1 (ω), ψ) − M (ψ) .

(5.61)

j=t

In order to verify Assumption 5.8 (iv), it suﬃces to show that for arbitrary but ﬁxed T > 0, ' ' sup 'S(t, n)'H = 0 P − a.s., (5.62) lim t→∞ t
where n(t, T ) is deﬁned in (5.48). To this end decompose (5.61) into S(t, n) = S1 (t, n) + S2 (t, n), where S1 (t, n) =

n−1 j=t

S2 (t, n) =

n−1

aj Mj (xj+1 (ω), Zj+1 (ω), ψ)

(5.63)

−E Mj (xj+1 (·), Zj+1 (·), ψ) , aj E Mj (xj+1 (·), Zj+1 (·), ψ) − M (ψ) .

(5.64)

j=t

Step 2. By assumption ,

G t, ·, x0 , Z0 , Υ (ψ) t∈N is a sequence of bounded functions which is L2 (Ω; RdH )-NED on {Dt }t∈Z of size − 21 , where Akk21 = σ(Dt | k1 ≤ t ≤ k2 ), k1 < k2 as before. By the minimummean-squared-error property and Assumption 5.15 (ii), we have ' '

'Mt (xt+1 (·), Zt+1 (·), ψ) − E Mt (xt+1 (·), Zt+1 (·), ψ)|At+k ' t−k L2 '

' ' ≤ 'Mt (xt+1 (·), Zt+1 (·), ψ) − Mt E xt+1 (·), Zt+1 (·) |At+k t−k , Ψ L2 '

' ' ≤ c4,t ' xt+1 (·), Zt+1 (·) − E xt+1 (·), Zt+1 (·) |At+k t−k L2 .

,

Hence for any ﬁxed ψ ∈ H with ψ H ≤ bH , Mt G t, ·, x0 , Z0 , Υ (ψ) , ψ t∈N is a sequence of functions which is L2 (Ω; H)-NED on {Dt }t∈Z of size −1/2. Now Lemma 5.3 implies that # "$

% (5.65) Mt (xt+1 (·), Zt+1 (·), ψ) − E Mt (xt+1 (·), Zt+1 (·), ψ) , At

5.5 Mathematical Appendix

121

with At = σ(Dk | − ∞ < k ≤ t) is an L2 (Ω; H) mixingale of size −1/2 with 1 magnitude indices dt = O(c4,t ) and decay parameters δt = O(k − 2 ). It follows from Lemma 5.2 together with Lemma 5.7 that for each α > 0, there exists N1 (α, ω) such that ' sup 'S1 (t, n) H ≤ α(1 + T ) P − a.s. t
for all t > N1 (α, ω). Step 3. Assumption 5.15 (ii) implies that for arbitrary α > 0, there exists N2 (α, ω) > N1 (α, ω) so that ' ' 'E Mt (xt+1 (·), Zt+1 (·), ψ) − M (ψ)' < α H for all t ≥ N2 (α, ω) and hence ' ' ' ' sup 'S2 (t, n)' ≤ αT t
H

P − a.s.

for all t ≥ N2 (α, ω). As a consequence ' ' sup 'S(t, n)'H ≤ α(2 + T ) P − a.s. t
for all t > N2 (α, ω). This establishes (5.62) and thus Assumption 5.8 (iv) is satisﬁed.

Proof of Proposition 5.4. For simplicity of notation, write t+k Et+k t−k [ · ] = E[ · |At−k ],

where Akk21 = σ(Ds | k1 ≤ s ≤ k2 ), k1 < k2 .

By Assumption 5.10 (ii) on the exogenous Rdξ -valued noise process {ξt }N , there exist λ > 21 and aξ > 0 such that −λ κξ (k) := sup ξt − Et+k , t−k [ξt ] L2 ≤ aξ k

k ∈ N.

(5.66)

t∈N

By construction, {G(t, ·, x0 , Z0 , Ψ )}t∈N is a sequence of RdH -valued bounded functions on (Ω, F , P). Thus for each k ∈ N, ' ' ' κG (k) := sup'G(t, ·, x0 , Z0 , Ψ ) − Et+k t−k [G(t, ·, x0 , Z0 , Ψ )] L < ∞. t∈N

2

In order to establish near-epoch dependency of {G(t, ·, x0 , Z0 , Ψ )}t∈N in the sense of Deﬁnition 5.4, we will show in two steps that κG (k) = O(k −λ ) with λ > 21 as given in (5.66). Step 1. Let (x0 , Z0 ) ∈ X × Σ, ω ∈ Ω be arbitrary, and Ψ ∈ S be a forecasting rule such that

122

5 Nonparametric Adaptive Learning

' ' e ' −λ κy (k) := sup'yte − Et+k , t−k [yt ] L2 ≤ ay k

k ∈ N,

t∈N

where

xt = G ξt , xt−1 , Ψ (xt−1 , Zt−1 ) . e = Ψ (xt−1 , Zt−1 ) yt−1

(5.67)

We show that the sequence of random variables {xt }t∈N generated by (5.67) is L2 (Ω; Rdx )-NED on {Dt }t∈Z of size − 21 so that ' ' −λ ' κx (k) := sup'xt − Et+k , t−k [xt ] L2 ≤ ax k

k∈N

(5.68)

t∈N

for some suitable constant ax > 0. The minimum-squared-error property of the conditional expectations operator and the triangle equality yield ' ' 'xt − Et+k [xt ]' t−k L2 '

' t+k t+k e e ' ≤ 'G(ξt , xt−1 , yt−1 ) − G Et+k t−k [ξt ], Et−k [xt−1 ], Et−k [yt−1 ] L2 ' ' ' ' t+k ' ' ' ≤ c1 'ξt − Et+k t−k [ξt ] L2 + c2 xt−1 − Et−k [xt−1 ] L2 ' ' e e ' + c3 'yt−1 − Et+k t−k [yt−1 ] L

2

' ' ' ' t+k−2i ' ' ≤ c1 'ξt − Et+k [xt−1 ]'L2 t−k [ξt ] L2 + c2 xt−1 − Et−k ' e ' e + c3 'yt−1 − Et+k−2i [yt−1 ]'L2 t−k

Taking suprema over t, this implies for each k ∈ N, κx (k) ≤ c2 κx (k − 1) + c3 κy (k − 1) + c1 κξ (k). We use induction over k to establish (5.68). Since c2 < 1, there exist δ > 0 with c2 (1 + δ) < 1 and k0 such that $ Set and observe that

k k−r

%λ

<1+δ

for all k > k0 .

" ax = max κx (k0 )k0λ ,

c3 (1+δ)ay +c1 aξ 1−c2 (1+δ)

κx (k0 ) ≤ ax k0−λ .

Suppose now that for some k > k0 , κx (k) ≤ ax k −λ . Then

# (5.69)

5.5 Mathematical Appendix

123

λ

λ

+ c3 ay k+1 + c1 aξ (k + 1)−λ κx (k + 1) ≤ c2 ax k+1 k k ≤ [c2 (1 + δ)ax + c3 (1 + δ)ay + c1 aξ ](k + 1)−λ ≤ ax (k + 1)−λ . Hence κx (k) = O(k −λ ) for all k ∈ N. Step 2. Since e e , xt−1 , . . . , yt−r , xt−r ), (xt , Zt ) = (xt , yt−1

we have '

' '(xt , Zt ) − Et+k (xt , Zt ) ' t−k L ≤

2

r r ' ' ' e ' 'y − Et+k [y e ]' 'xt−i − Et+k [xt−i ]' + t−i t−i t−k t−k L2 L2 i=0

≤

r

i=1 r ' ' e ' ' e 'yt−i − Et+k−2i [yt−i 'xt−i − Et+k−2i [xt−i ]' + ]'L . t−k t−k L 2

2

i=1

i=0

Taking the supremum over t, this implies κG (k) ≤ (ax + ay )

n r $ %λ k (k − i)−λ = (ax + ay ) k −λ = O(k −λ ). k−i i=0

i=0

This completes the proof of Proposition 5.4.

Proof of Lemma 5.5. Step 1. We ﬁrst prove the following lemma which is variant of Proposition 4.4 in Kuan & White (1994). Lemma 5.8. Assume that the hypotheses of Assumption 5.10 are satisﬁed. Let H : Ξ × Yn → Y, Y ⊂ Rdy be a Lipschitz continuous map such that ˜ ζ˜1 , . . . , ζ˜n ) ≤ c(1)

H(ξ, ζ1 , . . . , ζn ) − H(ξ, H

n

(2) ˜

ζi − ζ˜i + cH ξ − ξ

i=1 (1) for all ξ, ξ˜ ∈ Ξ, ζi , ζ˜i ∈ Y, i = 1, . . . , n. Suppose ncH < 1. Then the sequence of random variables {ζt }t∈N generated by

ζt = H(ξt , ζt−1 , . . . , ζt−n ),

t∈N

is L2 (Ω; Rdy )-NED on {Dt }t∈Z of size − 21 . In particular, for each δ > 0 there exists k0 such that ' '

−λ ' (5.70) κnζ (k) := sup'(ζt−1 , . . . , ζt−n ) − Et+k t−k (ζt−1 , . . . , ζt−n ) L2 ≤ ay k t∈N

for all k > k0 , where

124

5 Nonparametric Adaptive Learning

" ay := max κnζ (k0 )k0λ , . . . , κnζ (k0 − n + 1)(k0 − n + 1)λ , # (2) ncH (1+δ) a ξ (1)

(5.71)

1−ncH (1+δ)

with λ and aξ as given in (5.66). Proof of Lemma 5.8. The minimum-squared-error property of the conditional expectations operator and the triangle equality yield ' ' 'ζt − Et+k [ζt ]' t−k L 2

'

' t+k t+k ' ≤ 'H(ξt , ζt−1 , . . . , ζt−n ) − H Et+k t−k [ξt ], Et−k [ζt−1 ], . . . , Et−k [ζt−n ] L2 (1)

≤ cH

n ' ' ' ' 'ζt−i − Et+k [ζt−i ]' + c(2) 'ξt − Et+k [ξt ]' t−k t−k H L L 2

2

i=1 (1)

≤ cH

n

' ' (2) ' '

ζt−i − Et+k−2i [ζt−i ]'L2 + cH 'ξt − Et+k t−k t−k [ξt ] L2 .

i=1

Taking suprema over t, this implies for each k ∈ N, (1)

κζ (k) := sup ζt − Et+k t−k [ζt ] L2 ≤ cH t∈N

n

(2)

κζ (k − i) + cH κξ (k).

i=1

An induction argument over k is used next to establish (5.70). Choose δ > 0 (1) such that (1 + δ)n cH < 1 and k0 such that $

k k−n

%λ

<1+δ

for all k > k0 .

Set " b = max κζ (k0 )k0λ , . . . , κζ (k0 − n + 1)(k0 − n + 1)λ ,

#

(2)

cH

(1) 1−ncH (1+δ)

aξ

and observe that κζ (k0 − i) ≤ b(k0 − i)−λ ,

i = 0, . . . , n − 1.

Suppose now that for some k > k0 + 1, κζ (k − i) ≤ b (k − i)−λ ,

i = 1, . . . , n.

Then (1)

κζ (k) ≤ cH ≤

n

n $ %λ (2) (1) (2) k κζ (k0 − i) + cH κξ (k) ≤ cH b + cH aξ k −λ k−i

i=1 (1) [b ncH (1

i=1

+ δ) +

(2) cH aξ ]k −λ

≤ bk

−λ

.

5.5 Mathematical Appendix

125

This establishes near-epoch dependency of {ζt }t∈N . As regards the second assertion, we have n ' ' '

' '(ζt−1 , . . . , ζt−n ) − Et+k (ζt−1 , . . . , ζt−n ) ' ≤ 'ζt−i − Et+k [ζt−i ]' t−k t−k L L 2

2

i=1

≤

n ' ' 'ζt−i − Et+k−2i [ζt−i ]' t−k L2 i=1

Taking the supremum over t, this implies κnζ (k) ≤ b

n

(k − i)−λ = b

i=1

n $

k k−i

%λ

k −λ = (1 + δ)nb k −λ .

i=1

Setting ay := (1 + δ)nb, this completes the proof of Lemma 5.8. Step 2. In view of Remark 5.2, the forecasting rule for the m-th forecast is of the form e e e = Ψ (m) (xt , yt−1,t+1 , . . . , yt−1,t−1+m ). yt,t+m Assume for a moment that {xt }t∈N is near-epoch dependent so that (5.68) (1) (2) holds. We may then apply Lemma 5.8 with n = m − 1, cH = cH = cΥ , and {ξt }t∈N being replaced by {xt }t∈N to conclude that ' ' e ' −λ κy (k) = sup'yte − Et+k , k ∈ N, t−k [yt ] L2 ≤ ay k t∈N

where

" ay = max κy (k0 )k0λ , . . . , κy (k0 − n + 2)(k0 − n + 2)λ ,

(m−1)cΥ (1+δ) 1−(m−1)cΥ (1+δ) ax

#

(5.72) with λ and ax as given in (5.69). However, since {xt }t∈N and {yte }t∈N are interdependent processes generated by (5.67), ax and ay cannot be chosen independently. In fact, conditions (5.69) and (5.72) are two equations for two coeﬃcients ax and ay which must hold simultaneously. It is straightforward to ﬁnd ax and ay which fulﬁll conditions (5.69) and (5.72) if (m−1)cΥ (1+δ) c2 (1+δ) 1−(m−1)cΥ (1+δ) 1−c2 (1+δ)

<1

1 . Since δ may be chosen arbitrarily or, equivalently, if (m − 1)cΥ + c2 < 1+δ small, the latter condition is satisﬁed. This establishes near-epoch dependency

of {G(t, ·, x0 , Z0 , Ψ )}t∈N in the sense of Deﬁnition 5.4.

6 Stochastic Exchange Economies

As a ﬁrst application of the results obtained in Chapters 2 and 5, we discuss a standard version of the overlapping generations model with pure exchange which can be found in any macroeconomic textbook, e.g., see Azariadis (1993). The problem of learning the rational expectations equilibria when endowments are random has been treated by many authors, among which are Chen & White (1998) and Kelly & Shorish (2000). We will strengthen these results by showing that perfect foresight is possible despite of the fact that endowments are random. Generalizing ideas from Wenzelburger (2002) for the deterministic version of the model, we will introduce a learning scheme which generates strongly consistent nonparametric estimates of a so-called perfect forecasting rule which by deﬁnition generates perfect foresight orbits. We will show that the required structural knowledge is comprised in the error function which arises naturally when distinguishing between the basic market mechanism of the economy and a forecasting rule according to which agents form expectations. The zero contour set of an error function determines the so-called perfect forecasting rules which generate perfect foresight equilibria. Exploiting the geometry of the error function similar to Wenzelburger (2002), we develop an algorithm which generates an arbitrarily precise approximation of a perfect forecasting rule using historical data only. This algorithm converges globally for all initial conditions under fairly general conditions.

6.1 A Model of Pure Exchange Consider a standard version of the overlapping generations model with one non-storable commodity per period and ﬁat money as the only store of value between periods. There will be neither growth of the population nor production. Given the usual assumption of price taking behavior of all generations, young agents need to transfer purchasing power from the ﬁrst to the second period of their lives. To abstract from heterogenous beliefs, let pet denote the common forecast for the future price of the consumption good in period t + 1

128

6 Stochastic Exchange Economies

on which all members of the young generation in t base their decisions. We assume that a forecasting agency is in charge of issuing the forecasts and that this agency knows the basic market mechanism without information concerning the households’ savings behavior. (1) (2) Initial endowments are wt for young and wt for old consumers and are measured in units of the consumption good. They are assumed to be generated by a stationary stochastic process {wt }t∈N on a probability space (Ω, F , P) (1) (2) with values wt = (wt , wt ) in a compact cube W × W with W = [w, w]. At the beginning of each period t, a young agent observes her current endowment (1) (2) wt . The only uncertainty is over her future endowment wt+1 when old and the price level pt+1 prevailing in period t + 1. Consumers are endowed with a von-Neumann-Morgenstern utility function u : R2+ → R over consumption c(1) and c(2) when young and old, respectively. u is assumed to be bounded, strictly monotonically increasing, strictly concave and twice diﬀerentiable. The budget constraints are (1)

c(1) = wt

− s,

p c(2) = w(2) + s, q

where q ∈ R+ denotes the possible future price of the consumption good and w(2) the future endowment. Both are uncertain when deciding on how much to save. Let µ(pet , ·) ∈ Prob(R+ × W) denote the subjective joint distribution (2) of pt+1 and wt+1 , which is parameterized in subjective mean values pet . Given (1) the initial endowment of goods wt and the forecast pet for the unknown price level pt+1 , the solution of the consumer’s maximization problem is given by a savings function S : R3+ → R+ , deﬁned by 0 % $ (1) (1) e S(wt , p, pt ) := argmax u wt − s, w(2) + pq s µ(pet , dq, dw(2) ), (1)

0≤s≤wt (1) S(wt , p, pet )

such that at the goods price p.

R+ ×W

(6.1) describes real savings of young consumers in period t

Market clearing on the goods market in any period t requires that the real savings of young consumers, which deﬁnes the amount of consumption goods supplied to the market, has to be equal to the demand of the old generation which is equal to the real purchasing power of their money received in exchange for consumption goods in the previous period. If m ¯ denotes the total stock of money in the economy, a market clearing price in period t is deﬁned to be that price pt for which the excess demand of period t is equal to zero, i.e., $ % m ¯ (1) (1) − S wt , pt , pet = 0. Φex (wt , pt , pet ) := (6.2) pt Assume now that there exists an open subset Y ⊂ R+ and a map G : W×Y → R+ such that

6.2 The Existence of Perfect Forecasting Rules

Φex (w(1) , G(w(1) , pe ), pe ) = 0

129

for all (w(1) , pe ) ∈ W × Y.

If Φex is suﬃciently regular and satisﬁes the hypotheses of the implicit function theorem, a set Y together with a map G will exists. The market-clearing price pt of period t is then uniquely determined by the temporary equilibrium map G : W × Y → R+ ,

(1)

pt = G(wt , pet ).

(6.3)

The map G given in (6.3) includes a two-step-ahead forecast and for this reason deﬁnes an economic law with an expectational lead in the sense of Chapter 2. Since apart from the stochastic endowments only expected prices enter G as an argument, the economic law (6.3) has the structure of a Cobweb model. From a sequential view point, the forecast pet for the price pt+1 has to be determined before the actual trading takes place in period t and thus prior to the realization of pt . This feature is illustrated in Figure 6.1. t−1

t

t+1

wt−1

wt

pet−1

pet

pt−1

wt+1 pt

pet+1

pt+1

Fig. 6.1. Time-line of price formation.

It follows from (6.3) that the market-clearing price pt is a deterministic (1) function of the endowment wt and the price forecast pet . For this reason, we will assume that consumers probability distribution is of the form µ = δpet × ν, (2) where δpet is Dirac measure and ν is the probability distribution of wt+1 . As a consequence, the consumer’s savings function may assumed to be of the functional form 0 % $ $ % (1) pet (1) S wt , p := argmax u wt − s, w(2) + ppe s ν(dw(2) ). (6.4) (1)

0≤s≤wt

W

t

Remark 6.1. With the notation of Chapter 2 we would have written pet−1 ≡ pet−1,t+1 instead of pet . For the exchange model, however, the latter notation is economically more appealing.

6.2 The Existence of Perfect Forecasting Rules The discussion of the sequential trading structure in Section 6.1 revealed that perfect foresight may be possible despite the fact that endowments are generated by an exogenous stochastic process. Due to the particular information structure of the exchange economy, the mean error function associated with

130

6 Stochastic Exchange Economies

the economic law (6.3) becomes a deterministic error function of the form introduced in B¨ ohm & Wenzelburger (2004). In order to address the question whether forecasting rules exists that generate perfect foresight along all orbits (trajectories) of the economy, we therefore consider this deterministic error function. Analogously to the notion of an unbiased forecasting rule, a forecasting rule is called perfect if it generates perfect foresight orbits. Perfect forecasting rules are thus special cases of unbiased forecasting rules. (Compare with Remark 4.3, Chapter 2.) For an arbitrary period, let peold and penew denote the forecasts of the old and the young generation, respectively. The forecast error of the old generation is then given by the error function eG : W × Y2 → R, deﬁned by eG (w(1) , peold , penew ) := G(w(1) , penew ) − peold .

(6.5)

As in Chapter 2, the function eF describes all possible forecast errors, independently of what forecasting rule or learning scheme has been used to obtain the forecasts. The zero-contour set of eF describes all triplets (w(1) , peold , penew ) for which perfect foresight obtains. A perfect forecasting rule must therefore be a function penew = Ψ (w(1) , peold ) deﬁned on a suitable subset U ⊂ W × Y such that

(6.6) G w(1) , Ψ (w(1) , peold ) = peold . The economic implication for a perfect forecasting rule is that the new forecast penew has to be chosen such that the old forecast peold becomes correct. The following proposition shows that for the OLG exchange model forecasting rules which are perfect locally on some subset U exist under relatively mild conditions, whereas globally perfect forecasting rules exist only under very restrictive assumptions. Let ES (w(1) , θ) :=

D2 S(w(1) , θ)θ S(w(1) , θ)

denote the elasticity of the savings function with respect to expected inﬂation factors θ and S −1 (w(1) , ·), w(1) ∈ W denote the inverse of S(w(1) , ·), whenever this inverse is well deﬁned. Proposition 6.1. Let S be continuously diﬀerentiable, and assume that for each w(1) ∈ W, (i)

p¯w :=

m ¯ ∈Y (1) S(w , 1)

and

(ii)

D2 S(w(1) , 1) = 0.

Then there exists an open subset U ⊂ W × Y and a forecasting rule Ψ , given by ¯ (1) e e −1 (1) m Ψ (w , p ) = p S (6.7) w , e , (w(1) , pe ) ∈ U p which is locally perfect on U.

6.3 Perfect Foresight Dynamics

131

Proof. Let w(1) ∈ W be arbitrary but ﬁxed. Then Φex (w(1) , p¯w , p¯w ) =

m ¯ − S(w(1) , 1) = 0, p¯w

where Φex is the excess demand function deﬁned in (6.2). Since D2 S(w(1) , 1) = 0, D3 Φex (w(1) , p¯w , p¯w ) = 0. Invoking the Implicit Function Theorem, this yields the existence of an open set U (w(1) ) ⊂ Y such that

Φex w(1) , pe , Ψ (w(1) , pe ) = 0, pe ∈ U (w(1) ), where Ψ is given in (6.7). The deﬁnition of the economic law G deﬁned in (6.3), then shows

G w(1) , Ψ (w(1) , pe ) = pe , pe ∈ U (w(1) ). Since w(1) was arbitrary, the proof is completed by setting U = {(w(1) , pe ) ∈ W × Y| pe ∈ U (w(1) )}.

Corollary 6.1. Let youthful and old-age consumption be normal goods. Then D2 Ψ ≷ 0 on U if and only if D2 S ≶ 0 on U. Proof. We have

$ % ¯ D2 Ψ (w(1) , pe ) = S −1 w(1) , pm 1− e

1

¯ ES w(1) , S −1 (w(1) , pm e)

as long as D2 S = 0. The normality assumption together with the Slutsky conditions imply that ES (w(1) , θ) < 1. This completes the proof.

Proposition 6.1 assures that forecasting rules which are locally perfect will generically exist. The resulting dynamics under perfect foresight will be investigated in the next section.

6.3 Perfect Foresight Dynamics Proposition 6.1 states that perfect forecasting rules depend exclusively on previous forecasts and endowments but not on observed prices. Geometrically, the graph of a locally perfect forecasting rule Ψ is contained in the zero contour of the error function. Under perfect foresight in period t ∈ N, one has

(1) pt = G wt , pet = pet−1 , (6.8) (1) pet = Ψ (wt , pet−1 ),

132

6 Stochastic Exchange Economies (1)

where Ψ is the perfect forecasting rule deﬁned in (6.7) and wt ∈ [w, w] is random. Eq. (6.8) reveals that the dynamics under perfect foresight, if it exist, will essentially be governed by a dynamics on forecasts alone. (1) Since savings must never exceed initial endowments and wt is random, it follows from (6.2) and (6.7) that the map Ψ is not deﬁned for pt < m/w. ¯ Hence, each U must be a proper subset of W × R+ , even if Y = R+ . For this reason it is impossible to deﬁne a perfect foresight dynamics globally on R+ . In order to obtain a well-deﬁned perfect foresight dynamics, we seek a subset V ⊂ R+ with W × V ⊂ U which is forward-invariant under Ψ in the sense that Ψ : W × V −→ V. Then each orbit of (6.8) that starts in some pe0 ∈ V will stay in V for all times t. Proposition 6.2. Let youthful and old-age consumption be normal goods, let all hypotheses of Proposition 6.1 be satisﬁed, and assume that D2 S > 0. Then the compact interval V = [p, p] ⊂ R++ is forward-invariant under Ψ , if p p p S w, . ≤m ¯ ≤ p S w, p p Proof. We know from Corollary 6.1 that Ψ is monotonically decreasing in price forecasts, i.e. D2 Ψ < 0. Moreover, since both goods are normal we have D1 S > 0. This implies D1 Ψ > 0. Therefore, a compact interval [p, p] ⊂ R++ is forward-invariant under Ψ , if Ψ (w, p) ≥ p

and Ψ (w, p) ≤ p

By the monotonicity of S and the deﬁnition of Ψ , these conditions are equivalent to p m ¯ p m ¯ S w, and S w, ≥ ≤ . p p p p

If the hypotheses of Proposition 6.2 hold such that a forward-invariant set V = [p, p] exists, then one has perfect foresight along any orbit starting in V, that is, pt = pet−1 for all t ∈ N, provided that pe0 ∈ V. It follows from (6.8) that the resulting dynamics can be reduced to a one-dimensional one. In view of Assumption 4.1, Chapter 4 we assume that {wt }t∈Z has a representation as an ergodic metric dynamical system (Ω, F , P, {ϑt }t∈Z ) with wt (ω) = w(ϑt ω),

ω ∈ Ω, t ∈ Z.

(6.9)

Using the representation (6.9), we obtain a time-one map φ(1, ·) : Ω × V → V, given by

(6.10) pt+1 = φ 1, ϑt ω, pt := Ψ (w(1) (ϑt ω), pt ), of a random dynamical system over (Ω, F , P, {ϑt }t∈Z ). The time-one map (6.10) describes the price dynamics under perfect foresight of the system which

6.3 Perfect Foresight Dynamics

133

Ψ (w, ·)

Ψ (w, ·)

45◦

p

p0

p

Fig. 6.2. Orbit with perfect foresight.

is well deﬁned for all initial price levels p0 ∈ V = [p, p]. This situation is shown in Figure 6.2 which depicts a typical orbit of the system (6.10). We conclude this section by showing that under certain conditions, the long-run behavior is determined by a stationary stochastic process induced by a random ﬁxed point of the system (6.10). Theorem 6.1. Let the hypotheses of Proposition 6.1 be satisﬁed and assume that the savings function S is diﬀerentiable with respect to both arguments and D2 S > 0. Assume, in addition, that the following conditions hold: (i) There exits a compact set V = [p, p] which is forward-invariant under Ψ such that Ψ : W × V → V. (ii) There exists a random variable η with E[η] < 0, such that " #

sup log |D2 Ψ w(1) (ω), p | p ∈ V ≤ η(ω), ω ∈ Ω. Then there exists a unique nontrivial random ﬁxed point p : Ω → R+ which is asymptotically stable on V in the sense of Deﬁnition 4.6. Moreover, p is measurable with respect to the sub-σ-algebra of the past F − := σ(φ(t, ϑ−T ·, p) | 0 ≤ t ≤ T, p ∈ R+ ) ⊂ F, where φ denotes the random dynamical system generated by Ψ .

134

6 Stochastic Exchange Economies

The proof is an application of Theorem 4.3 in Section 4.4 of Chapter 4 and given in Section 6.5 below. It is a modiﬁcation of the proof of Theorem 4.2 in Schenk-Hopp´e & Schmalfuß (1998). Condition (ii) is an average contractivity condition. It is straightforward to see that (ii) is satisﬁed, if (ii )

|D2 Ψ (w(1) , p)| < 1 for all (w(1) , p) ∈ W × V.

Invoking Corollary 6.1, Condition (ii ) is satisﬁed at the deterministic ﬁxed points m ¯ p¯w = , w(1) ∈ W S(w(1) , 1) of the maps Ψ (w(1) , ·), w(1) ∈ W if ES (w(1) , 1) > 21 . It thus provides a natural extension of the classical stability condition for deterministic pure exchange models, cf. Azariadis (1993).

6.4 Adaptive Learning of Perfect Forecasting Rules Consider a forecasting agency which is in charge of forecasting the future evolution of the economy and hence has to decide on pe . We assume here that all young consumers in period t share the belief in pet issued by the forecasting agency prior to their savings decisions. This excludes heterogenous beliefs and strategic behavior of consumers. In order to be credible for households, we assume that the forecasting agency itself has no strategic interest other than issuing forecasts as precise as possible. The informational constraints faced by our forecasting agency can be described as follows. Suppose that the agency in some arbitrary period τ has observed past τ −1 τ −1 and knows forecasts {pet }t=0 corresponding to these prices. prices {pt }t=0 Recall that pτ is not available prior to the decision on peτ . Let the agency be aware of the basic market mechanism of the economy, that is, the basic structure of the economic law given by (6.3) but not its speciﬁc parameterization. Hence, neither the preferences nor the savings behavior of young consumers are known to the agency. The forecasting agency is assumed to use the concept of the error function without its exact functional speciﬁcation and for this reason is boundedly rational in the sense of Sargent (1993). The goal of a forecasting agency now is to estimate a perfect forecasting rule Ψ . Given the temporary equilibrium map (6.3), observe ﬁrst that in an arbitrary period τ , (1) pt = G(wt , pet ) τ −1 for all {(wt , pt , pet )}t=0 , no matter what forecasting rule has been applied to generate the time series. If a locally perfect forecasting rule Ψ exists, it (1 τ −1 must also satisfy follows from (6.6) that the data points {(wt , pt , pet )}t=0 (1

(1)

pet = Ψ (wt , pt )

(6.11)

6.4 Adaptive Learning of Perfect Forecasting Rules (1)

135

(1)

in period τ , provided that (wt , pt ) ∈ U. As wt , pt , and pet are observable quantities, the idea is to estimate Ψ nonparametrically from this data. With the notation of Chapter 5, let X = W × Y with Σ = Y ⊂ R+ and set (1)

xt := (wt , pt ) ∈ W × Y

and

Zt := pet ∈ Y.

Let 0 < cΥ < 1 be some constant, Y ⊂ Y be a compact interval not containing zero, and πY : R → Y be a projection onto Y . As in Section 5.5.1 of Chap1 ter 5, denote by CB (X; R) the space diﬀerentiable functions with bounded derivatives. Deﬁne the censor map as 1 1 CB (X; R) −→ CB (X; R) Υ : , (6.12) Ψ −→ Υ (Ψ ) where

Υ (Ψ ) =

and

πY ◦ Ψ % $ πY ◦ D2cΨΥ ∞ Ψ

if D2 Ψ ∞ < cΥ if D2 Ψ ∞ ≥ cΥ

" #

D2 Ψ ∞ := sup |D2 Ψ (w(1) , p)| (w(1) , p) ∈ W × Y .

The role of the censor map Υ is ﬁrst to conﬁne the system to a compact set Y at all stages and second to generate a near-epoch dependent series of forecasts as required by the convergence result of Theorem 5.3 in Chapter 5. The latter property follows from Lemma 5.8 of Chapter 5, noticing that for 1 (X; R) and each w(1) ∈ W, the map Υ (Ψ )(w(1) , ·) is a contraction. each Ψ ∈ CB Since X ⊂ R2+ , we choose the Sobolev space H = W2,2 (X; R) as introduced in Section 5.5.1 of Chapter 5 to be the underlying Hilbert space of our learning scheme. Recall that by Theorem 5.4, H can then be continuously imbedded into the space of continuous and bounded functions CB (X, R). For arbitrary (1) 1 initial conditions (w0 , p0 , pe0 ) ∈ X × Y and Ψˆ0 ∈ CB (X; R), our learning scheme is deﬁned by

(1) ⎧ pt = G wt , pet ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ pe = Ψt (w(1) , pe ) t t t−1 ⎪ ⎪ Ψt = Υ (Ψˆt ) ⎪ ⎪ ⎪ ⎩ˆ Ψt = Ψˆt−1 +

(6.13)

(1) 1 e ˆ t−1 Mt−1 (wt−1 , pt−1 , pt−1 , Ψt−1 )

with t ∈ N. Here the updating functions {Mt }t∈N are assumed to satisfy Assumption 5.15 of Chapter 5. For simplicity we suppressed the truncation device introduced in Section 5.1, knowing that under suitable conditions no truncations of the estimates will occur after a suﬃcient amount of time has elapsed. The next theorem shows under which conditions the learning scheme (6.13) succeeds in ﬁnding the perfect forecasting rule Ψ .

136

6 Stochastic Exchange Economies

Theorem 6.2. Consider an exchange economy as described above and assume that the following hypotheses are satisﬁed: (1)

(i) The stationary process {wt }t∈Z is L2 (Ω; R)-NED on {Dt }t∈Z of size − 21 , where {Dt }t∈Z is a sequence which is either uniformly mixing with αk of size −1 or strongly mixing with φk of size − 21 in the sense of Deﬁnition 5.3, Chapter 5. (ii) The economic law G when restricted to Y is Lipschitz continuous with respect to both variables. (iii) The conditions of Theorem 6.1 hold such that there exists a diﬀerentiable perfect forecasting rule Ψ ∈ H together with a forward-invariant set V ⊂ Y . Moreover, D2 Ψ ∞ ≤ cΥ < 1. (iv) The sequence {Mt }t∈N of updating functions satisﬁes Assumptions 5.125.15 of Chapter 5. 1 Let Ψˆ0 ∈ CB (X; R). Then the sequence of forecasting rules {Ψt }t∈N obtained from (6.13) converges to Ψ P-a.s., that is,

lim Ψt − Ψ H = 0

t→∞

P − a.s.

The proof of Theorem 6.2 is an application of Theorem 5.3, Chapter 5 and is found in Section 6.5 below.

Concluding Remarks The fact that perfect foresight is possible in a random environment of an exchange economy seems to have been overlooked in the literature. This chapter demonstrates that forecasting rules which generate perfect foresight exist under standard assumptions. Using the methodology of Chapter 5, these socalled perfect forecasting rules can successfully be estimated from historical data by means of nonparametric estimations. The censoring of the approximated forecasting rules assures that the system is kept stable at all stages of the estimation procedure and that the resulting process satisﬁes the necessary stochastic properties. It should be noted that the method required neither the existence of a perfect forecasting rule nor its contraction property.

6.5 Mathematical Appendix Proof of Theorem 6.1. Notice ﬁrst that by construction E[w(1) (·)] < ∞. We verify the assumptions of Theorem 4.3, given in Chapter 4. Step 1. Let V = [p, p]. Then V(ω) ≡ V, ω ∈ Ω is a random set and the set of all tempered random variables V associated with V is nonempty since it contains constant functions with values in [p, p].

6.5 Mathematical Appendix

137

Step 2. Check of Assumption (i). Condition (i) implies that V is forward invari- ant under Ψ . Since it is bounded by p, the map ω → Ψ ξ(ϑ−1 ω), g(ϑ−1 ω) is tempered, because it is bounded by a tempered (constant) map. Hence Assumption (i) is satisﬁed. Step 3. Condition (ii) is a mere restatement of Assumption (ii) in Theorem 4.3.

Step 4. Check of Assumption (iii). Let {φ t, ϑ−t ω, g(ϑ−1 ω) }t∈N be a Cauchy sequence for some g ∈ V and all ω, where φ denotes the random dynamical system generated by the time-one map Ψ . Then its limit limt→∞

φ(t, ϑ−t ω, g ϑ−1 ω) is in V for all ω, because V is forward invariant and complete. It remains to

show that this limit is contained in V. Since limt→∞ φ(t, ϑ−t ω, g ϑ−1 ω) is bounded by the constant p, it is bounded by a tempered random variable and hence must itself be tempered. It follows from steps 1-4 and Theorem 4.3 that there exists a unique random ﬁxed point p ∈ V whose domain of attraction contains the set V = [p, p]. Step 5. Measurability with respect to the past. Since V contains all constant functions g(ω) = p, p ∈ [p, p] and p is attracting on [p, p], the stationarity of ϑ implies that φ(T, ϑ−T ω, p) converges a.s. to p (ω) as T → ∞. Hence, p is measurable with respect to the sub-σ-algebra σ φ(t, ϑ−T ·, p)| 0 ≤ t ≤ T, p ∈ [p, p] ⊂ F.

Proof of Theorem 6.2. All Assumptions 5.10-5.15 of Chapter 5 are satisﬁed, so that Theorem 5.3 can be applied. Recall to this end that the censor map is a variant of the one introduced in Example 5.1, Chapter 5.

7 Heterogeneous Beliefs in a Financial Market

In this chapter the theoretical concepts of the ﬁrst chapters are applied to a ﬁnancial market with interacting agents. In recent years agent-based models have increasingly attracted attention in focusing on heterogeneous market participants as a central building block of a descriptive theory of ﬁnancial markets. It is a commonly accepted view that agents in ﬁnancial markets trade, since either preferences, beliefs or wealth positions diﬀer. Their attitudes towards risk and their beliefs about the future development of prices constitute the key inﬂuence on the determination of market prices. Behavioral assumptions concerning agents’ ability to foresee the future evolution of a market play a central role in modeling agent-based ﬁnancial markets. As traders, in general, will not share a common belief about future prices, the evolution of actual market prices cannot be self-conﬁrming for all of them. This is the reason why the issue of updating beliefs and learning in asset markets is of particular importance. Following an approach of B¨ ohm, Deutscher & Wenzelburger (2000) and B¨ohm & Chiarella (2005), we consider a model with heterogeneous myopic market participants characterized by preferences and subjective beliefs. Such a structure, for instance, arises naturally in models with overlapping generations of consumers. Consumers face investment possibilities in one risk-free and K ≥ 1 assets. They have no direct access to the ﬁnancial market and instead choose among ‘professional’ ﬁnancial mediators who carry out their transactions. We distinguish between two levels of rationality in the model. Each ﬁnancial mediator, who may be thought of as a stylized fund manager or broker, is boundedly rational in the sense of Sargent (1993) and is characterized by a forecasting technology which she uses to form expectations about future asset prices. On the contrary, consumers are boundedly rational in the sense of Simon (1982) and base their choice of a mediator on a performance indicator that measures the success of past investment strategies of the mediator. The imitating behavior of consumers is modeled by a standard discrete-choice approach as found in Anderson, de Palma & Thisse (1992). It constitutes the evolutionary feature of the model which is common

140

7 Heterogeneous Beliefs in a Financial Market

in many models including Brock & Hommes (1997a, 1998), Chiarella & He (2002, 2003a,b) or F¨ ollmer, Horst & Kirman (2005), Horst (2005) and many others. Given consumers’ preferences and the subjective beliefs of mediators, market-clearing prices at each date are determined by a temporary equilibrium map. Combined with forecasting rules that model the way in which mediators update beliefs, this yields a time-one map of a dynamical system in a random environment of the form discussed in Chapter 2 in which expectations feed back into the actual evolution of asset prices. The results of the present chapter are reﬁnements of results in Wenzelburger (2004b). Extending the concept of an unbiased forecasting rule of Chapter 2, we introduce perfect forecasting rules for second moments. These together with unbiased forecasting rules, also referred to as perfect forecasting rules for ﬁrst moments, provide correct ﬁrst and second moments of the price process conditional on all available information. In this sense, these perfect forecasting rules generate rational expectations for a particular mediator such as a fundamentalist while other market participants may have non-rational beliefs. We show that this concept derives naturally from the aggregate demand function of heterogeneous non-rational investors such as chartists and noise traders. Perfect forecasting rules for ﬁrst and second moments are required to obtain mean-variance eﬃcient portfolios for fundamentalists in the sense of classical CAPM theory. The natural question how a boundedly rational ﬁnancial mediator can learn perfect forecasting rules from historical data is addressed. The main informational constraint encountered by a mediator stems from the fact that neither the fraction of consumers choosing a particular mediator nor the beliefs of the mediators themselves are observable quantities. We will show that this missing information can be retrieved by estimating the excess demand function of all market participants. This will, in general, become a highly nonlinear problem. However, in the case when consumers are endowed with linear mean-variance preferences and chartists use simple linear forecasting rules, the main nonlinearity consists of the discrete-choice model which governs the switching behavior of consumers. Since rigorous techniques for estimating a discrete choice model exist (e.g., see Judge, Griﬃths, Hill, L¨ utkepohl & Lee 1985, Chap. 18), it turns out that the problem reduces to estimating a stochastic linear functional relationship. Using the methods introduced in Chapter 3, an adaptive learning scheme is developed which approximates perfect forecasting rules from historical data. Necessary and suﬃcient conditions for the convergence of the learning scheme are discussed.

7.1 The Model

141

7.1 The Model Consider an overlapping generations model with a ﬁnite number of types h = 1, . . . , H of young households, where each type consists of a total number of N (h) consumers, respectively. Each young household of type h lives for two periods, receives an initial endowment e(h) > 0 of a non-storable commodity in the ﬁrst period of life, and does not consume. In order to transfer wealth to the second period of his life, a consumer will choose a portfolio of K + 1 ∈ N retradeable assets whose proceeds he will consume. He receives no additional endowment in the second period of his life, so that his total consumption is equal to the return on the investment of his initial endowment when young. There is one risk-free asset which has a constant real rate of return R = 1 + r > 0 given exogenously, i.e., investing one unit of the consumption good in the risk-free asset yields R units of the consumption good in the subsequent period. The other K assets correspond to shares of ﬁrms whose production activities induce a stochastic process of dividends which are distributed to the shareholders. The total amount of retradeable risky assets per capita of young households is x ¯ ∈ RK +. Young consumers maximize expected utility over future consumption with respect to a von-Neumann-Morgenstern utility function and subjective expectations for future asset prices. Let y ∈ R denote the amount of the consumption good invested in the risk-free asset and x ∈ RK be the vector of shares purchased at the ex-dividend price vector p ∈ RK + (in units of the consumption good). Abstracting from short-sale constraints, the budget equation of a consumer of type h is e(h) = y + p x. Let d ∈ D denote a dividend pay K ment, which is randomly drawn from some subset D ⊂ RK + , p ∈ R+ be the future price vector of the risky assets, and q = p + d denote the corresponding cum-dividend price vector. Since there is no youthful consumption, the future wealth of consumer h associated with the portfolio x ∈ RK of risky assets is w(h) (x, p, q) = R (e(h) − p x) + risk-free investment

q x. equity return

Assuming that each consumer treats the price p at which he purchases as a parameter, the remaining uncertainty of the portfolio return rests with the cum-dividend price q of shares. Households have no direct assess to a forecasting technology and select a mediator who solves their individual investment problems using subjective probability distributions. There is a ﬁnite number i = 0, . . . , I of ﬁnancial mediators characterized by subjective beliefs regarding the future cum-dividend price of the assets. Their beliefs are described by subjective probability disK tributions ν (i) ∈ Prob(RK + ), i = 0, . . . , I, with Prob(R+ ) denoting the set K of all Borelian probability measures on R+ . For simplicity, we abstract from intermediation costs for households and suppress the constant e(h) whenever possible. The asset demand of a young consumer of type h based on the subjective belief ν (i) provided by mediator i is deﬁned by an optimal portfolio

142

7 Heterogeneous Beliefs in a Financial Market

choice which maximizes expected utility of future wealth 0

φ(h) (ν (i) , p) := argmax u(h) w(h) (x, p, q) ν (i) (dq). x∈RK

(7.1)

RK +

For the remainder of this chapter we will assume that the von-NeumannMorgenstern utility functions u(h) , h = 1, . . . , H and the subjective probability measures ν (i) , i = 0, . . . , I, are such that a unique solution (7.1) to the consumer’s decision problem exists. In particular, we assume that each distribution ν (i) , i = 0, . . . , I, is such that none of the assets is redundant. Existence and uniqueness can also be guaranteed by suitably restricting the class of utility functions and the class probability measures. This will be done in Deﬁnition 7.1 later in this chapter.1 We will express all aggregate quantities in terms of per capita of young households. Denote by N (h) η (h) := !H (h) h=1 N (hi)

the fraction of households of type h and ηt ∈ [0, η (h) ] be the fraction of !I (hi) households of type h employing mediator i in period t so that i=0 ηt = η (h) . Then H (i) (hi) ηt e(h) (7.2) Wt = h=1

is the amount of resources per capita which mediator i receives from all young households before trading takes place in that period. Her earnings from div(i) (i) idend and interest payments from the portfolio per capita (xt−1 , yt−1 ) ∈ (i) (i) RK × R obtained after trading in period t − 1 are d t xt−1 + ryt−1 . Since

aggregate repayment obligations to old households are (p + dt ) xt−1 + Ryt−1 , (i) her budget constraint in period t reads Wt = p x(i) + y (i) . Denote by (i) (1i) (Hi) ηt := (ηt , . . . , ηt ) ∈ [0, 1]H the distribution of households employing (i) mediator i in period t and νt denote mediator i’s subjective probability distribution for the future cum-dividend price qt+1 = pt+1 + dt+1 in period t. Then the aggregate demand function for risky assets per capita of all households which employ i in period t is (i)

(i)

(i)

x(i) = Φ(i) (ηt , νt , p) :=

H

(hi)

ηt

(i)

φ(h) (νt , p),

p ∈ RK +.

(i)

(7.3)

h=1

1

For further details we refer to the literature, e.g., see Grandmont (1982), Ingersoll (1987), and Pliska (1997).

7.1 The Model

143

Finally, denote by ξt ∈ RK a random quantity of assets per capita which noise traders hold after trading in period t.2 Following B¨ ohm, Deutscher & Wenzelburger (2000), the market-clearing asset price vector in period t is deﬁned by a solution pt ∈ RK + to I

! (i) (i) (i) Φ(i) (ηt , νt , p) − xt−1 + ξt − ξt−1 = 0.

(7.4)

i=0 (i)

(i)

If for a ﬁxed distribution of households ηt and ﬁxed beliefs νt the (percapita) aggregate excess demand for assets (7.4) is globally invertible with respect to p, then there exists a unique ex-dividend price pt which clears the asset market. This implies that the market-clearing asset price vector is a deterministic function of individual characteristics of young consumers and mediators. t−1 (i)

ηt−1

(i) νt−1

ξt−1 dt−1 pt−1

t

ξt (i)

ηt

(i) νt

dt pt

t+1 (i)

ηt+1

(i) νt+1

ξt+1 dt+1 dt+1

Fig. 7.1. Time-line of price formation.

In order to deﬁne a price map, note that the total amount of assets per !I (i) capita x ¯ must be equal to the sum of previous positions i=0 xt−1 + ξt−1 . Let (i) the market shares (ηt )Ii=0 ∈ ∆ with a suitably deﬁned simplex ∆ ⊂ [0, 1]HI (i) I and the subjective beliefs (νt )Ii=0 ∈ Prob(RK + ) of all the mediators at date t be given. Then under the above invertibility condition, the asset price vector which clears the market in period t is uniquely determined by the temporary I K equilibrium map G : RK × ∆ × Prob(RK + ) → R+ ,

(i) (i) pt = G ξt , (ηt , νt )Ii=0 .

(7.5)

The map (7.5) is an economic law in the sense of Chapter 2 extended to subjective and heterogeneous probability distributions. Observe that G does not contain the price itself as an argument, such that the map G is of Cobweb (i) type. The dating of the expectations νt relative to the actual price pt reveals that the economic law (7.5) has an expectational lead, that is, expectations are with respect to the realization of prices one period ahead of the map G. This observation is illustrated in Fig. 7.1. The functional form of (7.5) reﬂects 2

Noise traders are thought of speculators whose transactions are not captured by a standard microeconomic decision model. Alternative interpretations as given in De Long, Shleifer, Summers & Waldmann (1990, p. 709) apply as well.

144

7 Heterogeneous Beliefs in a Financial Market

the fact that asset prices are essentially determined by subjective beliefs of the ﬁnancial mediators and the way in which households choose among mediators. The decision of a household is based on the performance of a mediator. (i) The general idea is that the market shares of mediators (ηt )Ii=0 depend on the empirical distribution of the resulting price process and possibly other information. There are numerous ways of modeling this type of evolutionary behavior. A convenient way of formalizing this behavioral assumption is by means of a discrete choice approach (see Anderson, de Palma & Thisse 1992) and will be postponed to Section 7.4. For the next three sections, it suﬃces (0) (I) to assume that the market shares {(ηt , . . . , ηt )}t∈N are driven by some endogenous process which depends on empirical observations of the past. To describe the evolution of the asset prices, we need to specify the probabilistic assumptions on the exogenous noise and the exogenous dividend process. Assumption 7.1. Let (Ω, F , P) be a probability space and {Ft }t∈N an increasing family of sub-σ-algebras of F . Then we assume the following. (i) The dividend payments are described by a {Ft }t∈N -adapted stochastic process {dt }t∈N on (Ω, F , P) with values in D ⊂ RK + . The process is predictable meaning each dt , t ∈ N is Ft−1 measurable. (ii) The noise traders’ transactions are governed by a {Ft }t∈N -adapted stochastic process {ξt }t∈N on (Ω, F, P) with values in RK which is uncorrelated with the dividend process {dt }t∈N deﬁned in (i). Given the price law (7.5) for ex-dividend prices and Assumption 7.1 on the two exogenous processes, the cum-dividend price in period t is

(i) (i) qt = G ξt , (ηt , νt )Ii=0 + dt ,

t ∈ N.

(7.6)

Assumption 7.1 states that the dividend payment dt of period t is known whereas the noise trader transaction ξt is unknown before trading in period t. Mathematically this means that dt is assumed to be measurable with respect to Ft−1 , while ξt is assumed to be measurable with respect to Ft . Since the (i) (i) subjective beliefs (νt )Ii=0 and the market shares (ηt )Ii=0 must be set prior to trading in period t, they are based on information observable up to that time and thus must be Ft−1 measurable as well. This implies that the uncertainty in ex-dividend prices and in the traded quantities of assets rests solely on the behavior of the noise traders. In other words, the randomness of ex-dividend prices is essentially due to the stochastic nature of noise-trader behavior. Remark 7.1. The basic structure of the model follows B¨ ohm, Deutscher & Wenzelburger (2000) and B¨ ohm & Chiarella (2005). It should be emphasized that the underlying OLG structure is not essential for this setup. A generalization to multiperiod planning horizons is provided by Hillebrand (2003).

7.2 Heterogeneous Beliefs

145

7.2 Heterogeneous Beliefs This section is concerned with unbiased forecasting rules that generate correct ﬁrst moments of the price process along orbits of the system. The problem is central in agent-based ﬁnancial markets, since investors by their very nature have heterogeneous beliefs which they will not share. It has long been recognized that the two requirements of market clearing in all periods and rational expectations at all times can often not be fulﬁlled simultaneously. As indicated in Chapter 2 (see also B¨ ohm & Wenzelburger 2002), the evaluation of a forecasting rule can be carried out on several levels. For the purpose of the present model, we restrict our analysis to the comparison of the ﬁrst two moments of subjective and true distributions, i.e., on the conditional mean values and the conditional (variance-) covariance matrices. To simplify the analysis, assume that each mediator i = 0, . . . , I picks his subjective distribution ν (i) from a ﬁxed family of probability distributions parameterized in subjective mean values q (i) and subjective covariance matrices V (i) such that ν (i) ≡ νq(i) ,V (i) . Abstract from second moments beliefs (i)

and identify each subjective distribution νt at a particular date t with the (i) corresponding subjective mean value qt for the future cum-dividend price qt+1 = pt+1 + dt+1 . Let j be an arbitrary mediator and (−j)

qt

(0)

(j−1)

= (qt , . . . , qt

(j+1)

, qt

(I)

, . . . , qt ) ∈ RKI +

denote the respective subjective means of all mediators i = j. By abuse of notation, mediator j’s forecast error on the price qt prevailing at date t is

(i) (j) (−j) (j) (j) + dt − qt−1 , (7.7) qt − qt−1 = G ξt , (ηt )Ii=0 , qt , qt (j)

where the map G is given in (7.5) and qt−1 denotes the forecast for qt made at date t − 1. The functional form in (7.7) reveals that the forecast error for mediator j depends crucially on the current forecasts of all mediators. (i) Assume for a moment that the market shares (ηt )Ii=0 as well as the fore(−j) of the other mediators are known to mediator j. Since dt was casts qt assumed to be Ft−1 measurable, all uncertainty rests on the random transaction ξt of the noise-traders. Let Et−1 denote the expectations operator with respect to Ft−1 . In view of Chapter 2, all possible mean forecast errors on cum-dividend prices for mediator j at a particular date t can be described by a mean error function. In the context of this model, this mean error function (j) K(I+1) K is a map Et : ∆ × R+ × D × RK + −→ R , deﬁned by (j) (−j) (j) (j) (i) Et (ηt )Ii=0 , qt , qt , dt , qt−1

(7.8) (j) (j) (−j) (i) + dt − qt−1 , := Et−1 G ξt , (ηt )Ii=0 , qt , qt which describes all possible mean errors between actual cum-dividend prices and cum-dividend forecasts for mediator j conditional on information available prior to trading in period t. An unbiased forecasting rule for mediator j

146

7 Heterogeneous Beliefs in a Financial Market (j)

K K at date t, if it exists, is a map ψt : ∆ × RKI + × D × R+ → R+ with (j) (j) (i) (−j) (j) qt = ψt (ηt )Ii=0 , qt , dt , qt−1

(7.9)

such that (j)

Et

(i)

(j)

(ηt )Ii=0 , ψt

(i)

(−j)

(ηt )Ii=0 , qt

(j) (−j) (j) , dt , qt−1 , qt , dt , qt−1 = 0

K(I+1)

× D × RK identically on ∆ × R+ + . As shown in Chapter 4, Section 4.2, an unbiased forecasting rule (7.9) exists if the error function (7.8) satisﬁes the conditions of a global implicit function theorem. Recall that the implicit function theorem stipulates the particular functional form of (7.9). We will show below that an unbiased forecasting rule exists in the case in which all households have linear mean-variance preferences. An unbiased (j) forecasting rule for ex-dividend prices is obtained from (7.9) by setting pt = (j) qt − dt+1 as a forecast for pt+1 . It then follows from (7.8) that all forecast errors on ex-dividend prices for mediator j vanish identically, that is, Et [pt+1 ]− (j) pt = 0. In the sequel, an unbiased forecasting rule will also be referred to as a perfect forecasting rule for ﬁrst moments. The underlying economic intuition of an unbiased forecasting rule is as follows. By abuse of notation, write Φ(i) (η (i) , ν (i) , p) ≡ Φ(i) (η (i) , q (i) , p) for the respective aggregate demand functions (7.3), where we identify ν (i) with (j) q (i) as before. Assume, in addition, that for each p ∈ RK the + and each η (j) (j) (j) aggregate demand Φ (η , q , p) of mediator j is invertible with respect (j) (i) I to the price forecasts q (j) . Let qt−1 ∈ RK + and (ηt )i=0 be arbitrary and pt denote the market-clearing price in period t. The market-clearing condition (7.4) then implies that ex post, an unbiased forecasting rule (7.9) for j yielding (j) qt must satisfy

(j) (j) (i) (i) qt = Φ(j)−1 ηt , pt , x ¯ − ξt − Φ(i) (ηt , qt , pt ) . (7.10) i =j

The functional relationship (7.10) demonstrates that the essential unknown quantity for any mediator j is the excess demand of all other market participants. For any mediator, the existence of an unbiased forecasting rule thus depends exclusively on the fundamentals of the market mechanism, the subjective beliefs of all mediators who take part in the market, and on the way in which households choose their mediators. Consequently, any mediator with rational expectations has to be able to replicate the demand behavior of his fellow mediators. The main informational constraint is the fact that neither the market shares of a mediator nor her beliefs are observable quantities. The notion of a perfect forecasting rule for second moments, i.e., a forecasting rule which generates correct covariances matrices of the cum-dividend prices conditioned on the available information, could be derived using an analogous reasoning. We will postpone the discussion of such a rule to Section 7.5 when a more concrete example is available.

7.3 Risk Premia and Reference Portfolios

147

7.3 Risk Premia and Reference Portfolios In order to evaluate the performance of a mediator and thus the quality of households’ investment decisions, we analyze the risk premia obtained by mediators. We seek a portfolio which in the presence of diverse beliefs is (meanvariance) eﬃcient in the classical sense of CAPM. Naturally, such a portfolio must take into account discrepancies in beliefs. Throughout this section we assume that each conditional covariance matrix Vt [qt+1 ], t ∈ N is positive deﬁnite and hence invertible. For each t ∈ N deﬁne now the reference portfolio of period t by −1 [Et [qt+1 ] − Rpt ]. (7.11) xref t = Vt [qt+1 ] The reference portfolio xref of period t is a ‘ﬁctitious’ portfolio in the sense t that it is not necessarily held by a trader. It is determined after the realization of asset prices in period t. It is convenient to think of an investor who holds the reference portfolio after ﬁctitious trading in period t. Such a ﬁctitious investor may be an outside observer with linear mean-variance preferences with risk aversion 1 and rational expectations about future asset prices. Investing one unit of the consumption good into the risk-free and risky assets, gives the K ref portfolio (xref t , 1 − pt xt ) ∈ R × R. If r = R − 1 is the interest rate payed for the risk-free asset, then the return of the portfolio (7.11) obtained after a (ﬁctitious) selling at prices in period t + 1 is ref Rt+1 = r + [qt+1 − Rpt ] xref t .

(7.12)

ref ref The conditional variance Vt [Rt+1 ] of Rt+1 is given by ref Vt [Rt+1 ] = xref Vt [qt+1 ]xref t t

= Et [qt+1 ] − Rpt

(7.13)

Vt [qt+1 ]−1 Et [qt+1 ] − Rpt .

This implies that the (conditional) risk premium of the reference portfolio satisﬁes ref ref Et [Rt+1 ] − r = Vt [Rt+1 ]. (7.14) The reference portfolio of a ﬁnancial market is thus characterized by the property that its risk premium, also referred to as mean equity premium, is equal to its variance. For this reason, the risk premium (7.14) of the reference portfolio is always non-negative. (i) Let xt denote the portfolio held by mediator i in period t after investing (i) (i) (i) Wt given in (7.2). Then the realized return Rt+1 from selling xt in period t + 1 is (i) (i) Rt+1 = r + 1(i) [qt+1 − Rpt ] xt . (7.15) Wt

(i)

ref The conditional covariance Covt [Rt+1 , Rt+1 ] between the return of mediator (i) i’s portfolio xt and the return of the reference portfolio (7.11) is

148

7 Heterogeneous Beliefs in a Financial Market (i)

ref Covt [Rt+1 , Rt+1 ]=

1 (i) Wt

(i)

xt

Vt [qt+1 ]xref t .

(7.16)

Inserting (7.11) into (7.16), we see that the risk premium of mediator i’s portfolio takes the form (i)

(i)

ref Et [Rt+1 ] − r = Covt [Rt+1 , Rt+1 ].

(7.17)

Thus the risk premium of mediator i is equal to the covariance between i’s return and the return of the reference portfolio which, contrary to that of the reference portfolio, may well be negative. Combining (7.17) with (7.14) yields the following theorem. Theorem 7.1. Let i be an arbitrary mediator and assume that preferences and beliefs of all market participants are such that asset markets clear in each period t ∈ N at prices pt . Moreover, assume that Vt [qt+1 ] is positive deﬁnite for all t ∈ N such that the reference portfolio is well deﬁned. Then for each t ∈ N, (i) ref Covt [Rt+1 , Rt+1 ] (i) ref ]−r . Et [Rt+1 Et [Rt+1 ] − r = ref Vt [Rt+1 ] Theorem 7.1 states that the risk premium (7.17) of any mediator i can only be higher than the risk premium of the reference portfolio (7.11) at the (i) ref expense of higher risk, i.e. Vt [Rt+1 ] ≥ Vt [Rt+1 ]. Using the inequality 3 3 (i) (i) ref ref ], Covt [Rt+1 , Rt+1 ] ≤ Vt [Rt+1 ] Vt [Rt+1 Theorem 7.1 implies that for each mediator i = 0, . . . , I, the Sharpe ratios (conditional on information at date t) satisfy Et [Rt+1 ] − r Et [Rref ] − r 3 ref ] for all times t. 3 = Vt [Rt+1 ≤ 3 t+1 (i) ref Vt [Rt+1 ] Vt [Rt+1 ] (i)

(7.18)

As can be seen from (7.13), the upper bound in (7.18) is essentially determined by the endogenous price process. Using (7.15), the Sharpe ratios (7.18) of mediator i’s returns take the form (i)

(i) Et [Rt+1 ] − r Et [qt+1 ] − Rpt xt 3 = 3 , (7.19) (i) (i) (i) Vt [Rt+1 ] xt Vt [qt+1 ]xt (i)

showing that these are invariant under rescaling of the portfolio xt . Any (i) (i) (i) portfolio x ˜t = λ xt with λ > 0 will have the same Sharpe ratio as xt so that the size of a portfolio has no inﬂuence on the realized Sharpe ratios of a mediator.

7.4 Mean-Variance Preferences

149

Theorem 7.1 is a generalization of the famous security market line result (e.g., see LeRoy & Werner (2001) or Pliska 1997) to asset markets with heterogenous beliefs and preferences which are not necessarily of the meanvariance type. It demonstrates that in a world of heterogeneous investors, perfect forecasting rules for ﬁrst and second moments, if they exists, generate mean-variance eﬃcient portfolios in the sense of classical CAPM theory. Observe that the market-clearing condition in the proof of the theorem is not used so that the theorem easily generalizes to asset markets that do not necessarily clear such as electronic stock markets, cf. Li & Wenzelburger (2005).

7.4 Mean-Variance Preferences Starting with the work by Markowitz (1952) and Tobin (1958), mean-variance preferences provide a popular parameterization of preferences under risk. In the linear case, it is shown in B¨ ohm & Chiarella (2005) that the resulting aggregate demand function for shares is explicitly invertible. This allows for an explicit functional form of the market-clearing prices which greatly facilitates the analysis of adaptive learning. The relationship between mean-variance preferences and expected utility theory is given in the following deﬁnition. Deﬁnition 7.1. Let U : R × R+ −→ R denote a strictly concave function which is strictly increasing in its ﬁrst component and strictly decreasing in its second. U is said to represent mean-variance preferences if there exists a vonNeumann-Morgenstern utility function u : R −→ R and a class of probability measures P ⊂ Prob(R) such that 0 4

u(w )τ (dw ) = U Eτ [w], Vτ [w] , for all τ ∈ P. R

Here, Eτ and Vτ denote expected value and variance with respect to τ , respectively. (i)

As before, identify each subjective probability distribution νt on RK with (i) (i) its respective parameterization (qt , Vt ) and let w(h) (x, p, q) = Re(h) + (q − Rp) x denote future wealth of household h associated with the portfolio x ∈ RK .3 (i) (i) Then based on the belief (qt , Vt ), the subjective expected wealth of the K portfolio x ∈ R and its subjective variance conditional on information available prior to trading in period t are 3

The fact that the probability distributions are now deﬁned on the whole of R becomes a subtle technical necessity when linking expected utility theory with mean-variance preferences as done in Deﬁnition 7.1. K

150

7 Heterogeneous Beliefs in a Financial Market

Et [w(h) (x, p, ·)] = Re(h) + (qt − Rp) x (i)

(i)

and

Vt [w(h) (x, p, ·)] = x Vt x, (i)

(i)

respectively. Assume for the remainder of this chapter that each household h is characterized by linear mean-variance preferences, such that 3 (i) (i) U (h) Et [w(h) (x, p, ·)], Vt [w(h) (x, p, ·)] (i)

= Et [w(h) (x, p, ·)] −

α(h) (i) (h) (x, p, ·)], 2 Vt [w

describes the expected utility associated with the portfolio x ∈ RK in period t, where α(h) > 0 measures risk aversion. (hi) Given the market shares ηt of all mediators, the (per-capita) aggregate demand (7.3) of all households h which employ i is given by (i)

(i)

(i)

(i)

(i)−1

(i)

(7.20)

i = 0, . . . , I

(7.21)

Φ(i) (at , qt , Vt , p) = at Vt where (i)

at :=

H (hi) ηt , α(h) h=1

[qt − Rp],

denotes the risk-adjusted market share of the mediator i, i = 0, . . . , I. Set I −1 (i) (i)−1 At := at Vt (7.22) i=0

and note that At is well deﬁned and positive deﬁnite due to the positive def(i) initeness of all covariance matrices Vt . From the market-clearing condition (7.4), one immediately obtains an explicit functional form of the ex-dividend (i) (i) price law (7.5) which for arbitrary beliefs (qt , Vt )Ii=0 takes the form I

1 (i) (i) (i) (i) (i) I A q − At (¯ x − ξt ) (7.23) pt = G ξt , (at , qt , Vt )i=0 := R i=0 t t with

(i)

(i)

(i)−1

At := at At Vt

=

⎧ 0 ⎪ ⎪ ⎪ ⎨⎡ ⎪ ⎣ ⎪ ⎪ ⎩ IK +

(i)

a(j) t (i) j =i at

⎤−1 (i)

Vt

(j)−1 ⎦

Vt

if at = 0, (i)

(7.24)

if at > 0.

All coeﬃcient matrices (7.22) and (7.24) in the price law (7.23) are deter(i) mined by subjective covariances matrices Vt and risk-adjusted distributions of households (7.21) and hence are Ft−1 measurable.

7.4 Mean-Variance Preferences

151

The evolution of the market shares is modeled using a Logit model, see (hi) Anderson, de Palma & Thisse (1992).4 As before, let ηt denote the fraction of households h which employ mediator i and r = R − 1 be the interest rate payed for the risk-free asset. The return obtained by mediator i after trading in period t is (i)

(i)

Rt = r +

at−1 (i)

Wt−1

[qt − Rpt−1 ] Vt−1 [qt−1 − Rpt−1 ]. (i)−1

(i)

(7.25)

(i)

(i)

For each i, the sample mean µ ˆ t and the sample standard deviation σ ˆt (i) the time series {Rs }ts=0 are given by (i)

1 (i) R , t + 1 s=0 s 12 12 t t 1 (i) 1 (i) (i)2 := (R − µ ˆt )2 = R(i)2 − µ ˆt , t + 1 s=0 s t + 1 s=0 s

of

t

µ ˆt := (i) σ ˆt

(i)

(7.26)

(i)

ˆ−1 > 0. Using the empirical moments (7.26), an estimator where µ ˆ−1 ≥ 0 and σ for the Sharpe ratio associated with the realized returns (7.25) of mediator i (i) (i) is given by (ˆ µt − r)/ˆ σt . Recall that the fraction of households of type h is (hj) η (h) . The fraction ηt of households of type h which employs mediator j in period t is now assumed to be determined by the discrete-choice probability $ % (j) (j) exp β (h) (ˆ µt−1 − r)/ˆ σt−1 (hj) $ % , t ≥ 0. := η (h) ! ηt (7.27) (i) (i) I (h) (ˆ µt−1 − r)/ˆ σt−1 i=0 exp β The parameter β (h) appearing in the modiﬁed logit function (7.27) describes the intensity of choice of a household of type h, that is, how fast a typical consumer of type h will switch to a diﬀerent mediator. The reference portfolio takes the following form. Inserting (7.23) into (7.11) !I (i) and using IK = i=0 At for all times t, we obtain −1 xref At (¯ x − ξt ) + t = Vt [qt+1 ]

I

(i)

Vt [qt+1 ]−1 At

(i)

Et [qt+1 ] − qt

.

(7.28)

i=0

Formula (7.28) allows us to interpret the reference portfolio (7.11) as a ‘modiﬁed’ market portfolio that accounts for discrepancies in incorrect beliefs. Indeed, if all mediators had correct ﬁrst and second conditional moments of the 4 There are numerous ways of modeling the switching behavior of boundedly rational households, see Brock & Hommes (1998). We will follow the intuition that boundedly rational consumers with mean-variance preferences would prefer portfolios with the highest empirical Sharpe ratio because such portfolios promise to be mean-variance eﬃcient in the sense of the classical CAPM theory.

152

7 Heterogeneous Beliefs in a Financial Market (i)

(i)

price process, that is, Et [qt+1 ] = qt and Vt [qt+1 ] = Vt then 1 xref (¯ x − ξt ). t = ! (h)

for all i = 0, . . . , I,

H η h=1 α(h)

This shows that the reference portfolio (7.11) is collinear to the (per-capita) market portfolio x¯ if all mediators have rational expectations and noise traders are absent. Remark 7.2. The price law (7.23) as well as the resulting price process would be the same for consumers with inﬁnite lives who maximize wealth myopically. The setup can thus be seen as an extension of Brock & Hommes (1997a, 1998) and Chiarella & He (2002) to H types of households, I mediators, and K risky assets. A generalization of this setup to multiperiod planning horizons is provided in Hillebrand & Wenzelburger (2006a,b).

7.5 Perfect Forecasting Rules In this section we investigate the existence of forecasting rules which generate rational expectations on cum-dividend prices for mediator 0 in the sense that the ﬁrst two moments of the future asset prices are correctly predicted. In view of mediators who seek mean-variance eﬃcient portfolios, Section 7.3 showed that it suﬃces to analyze the case in which the ﬁrst two moments of subjective and true distributions, i.e., the conditional mean values and the conditional covariance matrices, coincide. For brevity we will use the term rational expectations to describe the situation in which a boundedly rational mediator is able to correctly predict the ﬁrst two moments of the price process conditional on all available information, whereas other market participants may have non-rational beliefs. (0) Consider mediator 0 and assume without loss of generality that at > 0 for all times t ∈ N throughout this section. This can always be guaranteed by (h 0) setting η0 > 0 and β (h ) = 0 for some household h in the logit functions (0) (7.27). On the other hand at = 0 imposes no restriction on the rationality of 0 because it simply means that no household employs mediator 0 such that her beliefs do not feed back into the price process. In this case there is no existence problem and the perfect forecasting rules simply coincide with the ﬁrst two moments of the corresponding price process. We will ﬁrst address the existence of perfect forecasting rules for ﬁrst moments and then the existence of perfect forecasting rules for second moments. 7.5.1 Perfect Forecasting Rules for First Moments Recalling that dt is assumed to be Ft−1 measurable, the expected cumdividend price conditioned on information available at date t associated with the price law (7.23) is

7.5 Perfect Forecasting Rules

I 1 (i) (i) (0) (0) A q + At qt − At (¯ x − Et−1 [ξt ]) + dt . Et−1 [qt ] = R i=1 t t

153

(7.29)

The condition that the forecast errors for mediator 0 vanish in the mean is (0)

Et−1 [qt − qt−1 ] = 0

(7.30)

for all times t. Inserting (7.29) into (7.30) and rearranging, yields an explicit (0) expression for the new forecast qt , given by I (i) (i) (0) (0)−1 (0) At qt + At (¯ x − Et−1 [ξt ]) . (7.31) qt = At R(qt−1 − dt ) − i=1

Replacing the coeﬃcients in (7.31) with (7.24), the unbiased forecasting rule for mediator 0 takes the form

(0) (i) (i) (i) (0) (7.32) qt = ψ (0) Et−1 [ξt ], dt , (at , qt , Vt )Ii=1 , qt−1 (0)

:= R(qt−1 − dt ) (0)−1 (0) Vt − at

I

(i) (i)−1 (i) qt at Vt

−

(0) R(qt−1

− dt ) + Et−1 [ξt ] − x ¯

.

i=1

The functional form of the unbiased forecasting rule (7.32), also referred to as a perfect forecasting rule for ﬁrst moments, is closely related to that of (7.10) in Section 7.2. The quantity in (7.32) which a priori is unknown to mediator (0) 0 is the term in the large brackets. Since by (7.30) Et−1 [pt ] = qt−1 − dt , this term describes precisely the expected excess demand of all mediators i > 0 including the expected demand of the noise traders Et−1 [ξt ]. The functional form (7.32) of the unbiased forecasting rule thus reﬂects the intuition that the precision of forecasts requires precise knowledge of the investment behavior of all market participants. 7.5.2 Perfect Forecasting Rules for Second Moments While in the case of linear mean-variance preferences perfect forecasting rules for ﬁrst moments always exist, stricter requirements for forecasting rules which generate correct second moments of the price process are needed. Such rules will be referred to as perfect forecasting rules for second moments. Observe (0) that the subjective covariance matrix Vt in the expression for (7.32) was until now arbitrary. We will now try to determine this matrix as to provide correct second moments of the price process. Since by Assumption 7.1 (ii) ξt and dt are uncorrelated, the conditional covariance matrix of the cum-dividend prices are obtained from the price law (7.23) and (7.29), yielding

154

7 Heterogeneous Beliefs in a Financial Market

1 (i) (i) (i) (7.33) Vt−1 [qt ] = Vt−1 G ξt , (at , qt , Vt )Ii=0 = 2 At Vt−1 [ξt ]At . R $! %−1 (i) (i)−1 I Recall that At = a V is positive deﬁnite such that Vt−1 [qt ] t i=0 t is invertible if Vt−1 [ξt ] is non-degenerate. This implies that the variance and the covariance of the price process receives an expectations feedback from the (i) subjective covariance matrices (Vt )Ii=0 as long as there is noise in the system, i.e., Vt−1 [ξt ] is diﬀerent from the zero matrix. The volatility in the price process (7.33) is therefore exclusively generated by the noise-trader behavior, the switching behavior of households, and the subjective covariance matrices of the mediators. The condition that mediator 0’s forecast errors for second moments (7.33) of the price process vanish for all times t is (0)

Vt−1 [qt ] − Vt−1 =

1 (0) ! At Vt−1 [ξt ]At − Vt−1 = 0. R2

(7.34)

The idea for a perfect forecasting rule for second moments is analogous to the (0) idea for perfect rules for ﬁrst moments: choose a new forecast Vt such that (0) (7.34) holds and the old forecast Vt−1 becomes correct. Inserting (7.22) into (7.34) yields the necessary condition

I i=0

−1 (i) (i)−1 at Vt

Vt−1 [ξt ]

I

−1 (i) (i)−1 at Vt

(0)

= R2 Vt−1

(7.35)

i=0 (0)

for the subjective covariance matrix Vt . As a consequence, perfect forecasting rules for second moments are determined by symmetric positive deﬁnite (0) solutions Vt to (7.35). The existence proof is divided into two steps. First ﬁnd a symmetric pos(0) itive deﬁnite matrix Πt that solves the equation Πt−1 Vt−1 [ξt ]Πt−1 = R2 Vt−1 . Then set −1 I (i) (i)−1 (0) (0) . Vt = at at Vt Πt − i=1

If this matrix is positive deﬁnite, then the desired second moment is found. In order to state the conditions under which this is the case and a perfect forecasting rule for second moments exists, we deﬁne the square root of a symmetric positive deﬁnite matrix. Recall that any symmetric and positive deﬁnite K × K matrix B can be diagonalized so that B = O diag(λ1 , . . . , λK )O for real eigenvalues λ1 , . . . , λK√> 0 and for some orthogonal matrix O, i.e. B of B is√a symmetric positive deﬁnite O O = IK . The √ square root √ √ √ matrix, deﬁned by B := O diag( λ1 , . . . , λK )O, such that B = B B. We summarize these ﬁndings in the following proposition. All technical details of the proof are found in the mathematical appendix, Section 7.8.

7.5 Perfect Forecasting Rules

155

Proposition 7.1. Let the hypotheses of Assumption 7.1 be satisﬁed and suppose that each Λt−1 := Vt−1 [ξt ], t ∈ N is positive deﬁnite. Set 5$ %−1 4 4 4 4 (0) 1 Πt = R Λt−1 Λt−1 Vt−1 Λt−1 Λt−1 , t ∈ N. (7.36) Then the forecasting rule for second moments ϕ(0) , given by (0)

Vt

(i) (i) (0) (0) = ϕ(0) Λt−1 , (at , Vt )Ii=1 , at , Vt−1 :=

(0) at

Πt −

I

(7.37)

−1 (i) (i)−1 at Vt

,

i=1

provides correct second moments of the price process at date t in the sense that !I (i) (i)−1 (0) is positive deﬁnite and at > 0. (7.34) holds whenever Πt − i=1 at Vt A special case occurs when noise traders induce zero correlation3between (0)−1 σ diﬀerent assets and Vt−1 [ξt ] ≡ σξ2 IK . Then Πt reduces to Πt = Rξ Vt−1 . If the r.h.s. of (7.37) is neither well deﬁned nor positive deﬁnite, the resulting expression on the left deﬁnes no covariance matrix such that a forecasting rule (7.37) which is perfect for all times t may fail to exist. Hence, additional requirements which guarantee that (7.37) is well deﬁned for all periods are needed. There are two special cases in which perfect forecasting rules for second moments exists. In the ﬁrst one, all mediators agree upon subjective second moments. In the second case, all mediators i > 0 believe in constant covariance matrices. Lemma 7.1 establishes existence of perfect forecasting rules for second moments in the ﬁrst case, Lemma 7.2 in the second case. Both proofs are found in the mathematical appendix, Section 7.8. (i)

(0)

Lemma 7.1. Under the hypotheses of Proposition 7.1, suppose Vt ≡ Vt for all i = 1, . . . , I and all times t. Then the forecasting rule (7.37) takes the form H 5$ %3 η (h) 3 4 (0) (0) 4 −1 Λ Λ Λ Λ−1 Vt = R V t−1 t−1 t−1 t−1 . t−1 (h) α h=1 If, in addition, Vt−1 [ξt ] ≡ Λ is constant over time, then the constant rule (0) Vt

≡

H η (h) R α(h) h=1

2

is perfect for second moments as well.

Λ−1

for all t ∈ N

156

7 Heterogeneous Beliefs in a Financial Market

Lemma 7.2. Under the hypotheses of Assumption 7.1, let Vt−1 [ξt ] ≡ σξ2 IK with σξ > 0. Assume that the following hypotheses are satisﬁed. (i)

(i) All mediators i > 0 use constant covariance matrices Vt ≡ V (i) such that V (i)−1 = O diag(λ(i1) , . . . , λ(iK) ) O, i = 1, . . . , I for some orthogonal K × K matrix O. (ii) The eigenvalues λ(i1) , . . . , λ(iK) , i = 1, . . . , I given in (i) satisfy H 6 7 I η (h) 7 σξ 8 λ(ik) < , k = 1, . . . , K. 2R α(h) i=1 h=1

(iii) The initial forecast is V0 := O diag(λ0 (01) (0K) given in (i) and λ0 , . . . , λ0 > 0. (0)

(01)

(01)−1

(0K)−1

, . . . , λ0

)O, where O is

(0K)

are suitably large, then the forecasting If the initial eigenvalues λ0 , . . . , λ0 (0) rule (7.37) is perfect for second moments whenever at > 0. Observe that perfect forecasting rules for second moments are not necessarily uniquely determined. This is illustrated by Lemma 7.1. Mathematically this is due to the natural appearance of multiplicities in matrix polynomial equations, cf. Gohberg, Lancaster & Rodman (1982). Of course, this multiplicity will already appear in the one-asset case K = 1 when (7.34) reduces to a scalar quadratic equation. Setting aside the existence problem, economically the main informational constraint for applying the forecasting rule (7.37) is again the fact that neither the market shares of a mediator nor the subjective covariances matrices of the other mediators are observable quantities. We will show in Section 7.7 how this missing information may be estimated from historical data, such that a suitable approximation of (7.37) may be used whenever it is well deﬁned.

7.6 Dynamics Under Rational Expectations In this section we discuss the dynamics of asset prices when, under the hypotheses of Section 7.5, mediator 0 has rational expectations in the sense that ﬁrst moments of cum-dividend prices are correctly predicted for all times t and second moments whenever possible. Assuming that each mediator i > 0 uses some forecasting rules ψ (i) and ϕ(i) , the prices under rational expectations for group 0 are determined by

7.6 Dynamics Under Rational Expectations

⎧ qt ⎪ ⎪ ⎪ ⎪ ⎪ (i) ⎪ ⎪ qt ⎪ ⎪ ⎨ (i) Vt ⎪ ⎪ ⎪ ⎪ (0) ⎪ qt ⎪ ⎪ ⎪ ⎪ ⎩ (0) Vt

(0)

= qt−1 +

1 R At (ξt

157

− Et−1 [ξt ]),

= ψ (i) (qt−1 , . . . , qt−L ),

i = 1, . . . , I,

= ϕ(i) (qt−1 , . . . , qt−L ), i = 1, . . . , I,

(i) (i) (i) (0) = ψ (0) Et−1 [ξt ], dt , (at , qt , Vt )Ii=1 , qt−1 ,

(i) (i) (0) = ϕ(0) Vt−1 [ξt ], (at , Vt )Ii=1 , Vt−1 ,

(7.38)

where ϕ(0) coincides with (7.37) whenever (7.37) is well deﬁned. The cumdividend price qt in (7.38) is essentially determined by the forecasting rule ψ (0) of mediator 0 which takes the form : 9 (i) (i) ! ! (0) (i)−1 (0) (0) (i)−1 (i) (0) at at qt = R IK + Ii=0 (0) qt−1 − Ii=1 (0) Vt Vt Vt Vt qt at at (7.39) ! (i) (i)−1 (0)−1 (0) R Ii=0 at Vt dt + Et−1 [ξt ] − x ¯ . − at Vt The map (7.38) together with the logit functions (7.27) with (7.21) deﬁnes a time-one map of a dynamical system in a random environment which describes the evolution of prices. The map (7.38) will, in general, be non-linear because nonlinearities may enter the system through forecasting rules ϕ(i) and ψ (i) , i > 0. Since time appears as an explicit argument in the discrete choice model, the map (7.38) is not a random dynamical system in the strict sense of Arnold (1998). It is beyond the scope of this chapter to conduct a stochastic analysis of the system (7.38) in the spirit of F¨ ollmer, Horst & Kirman (2005) and Horst (2005) or to present a dynamic investigation paralleling numerous works initiated by Brock & Hommes (1998). The interested reader is referred to Horst & Wenzelburger (2005). For the remainder of this section, we instead discuss two cases of a stochastic linear benchmark dynamics. Assume for this (i) purpose that market shares are constant over time, that is at ≡ a(i) , i = 0, 1. This case, for instance, obtains if the intensity of choice parameters in (7.27) are all set to zero and no household switches between mediators.5 Case I. Assume that there are only two mediators i = 0, 1. Let mediator 1 be a chartist who uses the simple technical trading rule (1)

qt

= ψ (1) (qt−1 , . . . , qt−L ) :=

L

D(l) qt−l

(7.40)

l=1 (l)

(l)

as a forecasting rule, where D(l) = diag(δ1 , . . . , δK ), l = 1, . . . , L are diagonal matrices whose non-zero entries are weighted trends of asset prices, 5

Numerical simulations of the model indicate that the market shares settle relatively fast to constant values, at least for suﬃciently small parameters of the intensity of choice. As switching becomes negligible in the long run, the system then is asymptotically linear, cf. B¨ ohm & Wenzelburger (2005).

158

7 Heterogeneous Beliefs in a Financial Market

respectively. In addition, let all subjective covariance matrices be constant (i) over time, i.e. Vt ≡ V (i) , i = 0, 1. Then, under the hypotheses of Lemma 7.2, mediator 0 may have correct second moments which are constant over (0) K(L+1) detime, since no switching occurs. If Qt = (qt , qt , . . . , qt+1−L ) ∈ R+ notes the vector consisting of the last price forecast of mediator 0 and past L realized prices, then (7.38) deﬁnes an ARMAX process Qt = AQt−1 + bt

(7.41)

with a constant coeﬃcient block matrix A and a vector-valued random variable bt . More precisely, setting A0 := R(IK +

a(1) (0) (1)−1 V V ), a(0)

Al :=

⎛

we have

−a(1) (0) (1)−1 (l) V V D , a(0)

l = 1, . . . , L,

⎞ · · · AL ··· 0 ⎟ ⎟ , . . .. ⎟ . . ⎠ 0 · · · IK 0

A0 ⎜ IK ⎜ A := ⎜ . ⎝ ..

··· 0 .. .

(7.42)

whereas ⎛ ⎜ ⎜ ⎜ ⎜ bt = ⎜ ⎜ ⎜ ⎝

⎞ x ¯ − Et−1 [ξt ] − R[a(0) V (0)−1 + a(1) V (1)−1 ] dt ⎟ −1 ⎟ 1 (0) (0)−1 (1) (1)−1 ⎟ V + a V (ξ − E [ξ ]) a t t−1 t R ⎟ ⎟. 0 ⎟ ⎟ .. ⎠ .

1 V (0) a(0)

(7.43)

0 In view of Chapter 4, assume that each of the two process {ξt }t∈N and {dt }t∈N can be extended to processes on Z and represented by an ergodic metric dynamical system (Ω, F , P, {ϑt }t∈Z ). Then the stochastic process (7.43) can be written as bt (ω) = b(ϑt ω) with some measurable map b : Ω → RK(L+1) and (7.41) becomes an aﬃne random dynamical system in the sense of Arnold (1998, Chap. 5). A stationary solution to the stochastic diﬀerence equation (7.41) is given by a so-called6 random ﬁxed point Q (ω) which is deﬁned by the random variable Q (ω) =

∞

As b(ϑ−(s+1) ω),

ω ∈ Ω.

(7.44)

s=0 6 In classical terminology, a random ﬁxed point of (7.41) corresponds to a ‘steady state solution’ of a linear stochastic system, cf. Hannan & Deistler (1988, Chap. 1). An asymptotically stable random ﬁxed point with rational expectations is also a special case of a stochastic consistent expectations equilibria in the sense of Hommes, Sorger & Wagner (2004), see Chapter 4, Section 4.4.

7.6 Dynamics Under Rational Expectations

159

Under certain conditions given in the next proposition, the random ﬁxed point Q (ω) is said to be globally asymptotically stable, implying that P-almost all solutions to equation (7.41) will eventually behave like the stationary stochastic process {Q (ϑt ω)}t∈N induced by (7.44). Here, the stationarity follows from the stationarity assumption for the exogenous noise processes. Proposition 7.2. Assume that the following hypotheses are satisﬁed. (i) {ξt }t∈N and {dt }t∈N can be represented by an ergodic metric dynamical system (Ω, F, P, {ϑt }t∈Z ), where E[ξt ] ≡ ξ ∈ RK and E[dt ] ≡ d ∈ RK +. (i) (i) (i) (i) (ii) Vt ≡ V and at ≡ a > 0, i = 0, 1. (iii) det(A − λI) = 0 for all λ ∈ C with |λ| ≥ 1, that is, all eigenvalues of A given in (7.42) lie inside the unit disk. Then the random ﬁxed point Q (ω) deﬁned in (7.44) is well deﬁned, uniquely determined, and globally asymptotically stable. The mean price level q ∈ RK + is the uniquely determined solution of the linear equation (I − A)Q = E[b], where Q := (q, . . . , q) ∈ RK(L+1) . Proof. The stability property follows from Arnold (1998, Corollary 5.6.6). The second statement is due to the fact that I − A is invertible such that (I − A)E[Q ] = E[b] has a unique solution of the form E[Q ] = (q, . . . , q). Case II. Another special case arises when the two mediators agree on the (0) (1) subjective covariance matrices, such that Vt ≡ Vt for all times t. This (i) implies that all At , i = 0, 1 are time-invariant diagonal matrices and the map (7.39) takes the form % $ (1) (1) !L (0) (0) qt = R 1 + aa(0) qt−1 − aa(0) l=1 D(l) qt−l % $ (7.45) (1) (0) +a(0)−1 Vt (¯ x − Et−1 [ξt ]) − R 1 + aa(0) dt . Since all D(l) , l = 1, . . . , L are diagonal matrices, (7.45) decouples into K (0) scalar equations which are linked by an inhomogeneous term via Vt . The stability properties of (7.38) are now obtained from an analysis of each scalar equation for each asset k = 1, . . . , K. Proposition 7.3. Assume that the following hypotheses are satisﬁed. (i) Hypothesis (i) of Proposition 7.2 holds, where, in addition, Vt−1 [ξt ] ≡ Λ is constant over time. (i) (0) (1) (ii) at ≡ a(i) > 0, i = 0, 1 and Vt ≡ Vt for all t ∈ N. Moreover, each (0) Vt , t ∈ N is obtained from a constant perfect forecasting rule for second %2 $ ! (h) (0) H moments as given by Lemma 7.1, such that Vt ≡ R h=1 αη (h) Λ−1 .

160

7 Heterogeneous Beliefs in a Financial Market

(iii) Let 1 ≤ k ≤ K be arbitrary but ﬁxed. Assume that all zeros of the characteristic polynomial χk associated with (7.45), which is given by $ χk (λ) = λL+1 − R 1 +

a(1) a(0)

%

λL +

a(1) a(0)

L

(l)

δk λL−l ,

l=1

lie inside the unit circle. Then the price process for the k-th asset admits a globally asymptotically stable random ﬁxed point. The mean price level of the k-th asset q k is given by qk =

a(1) a(0) +a(1)

yk − Rdk a(0) +a(1) $! % (l) L l=1 δk − 1

−R−1

$ ! H where yk is the k-th entry of the vector y = R h=1

η (h) α(h)

, %2

Λ−1 (¯ x − ξ) ∈ RK

and dk is the mean value of the dividend payment of the k-th asset. The proof of Proposition 7.3 is analogous to that of Proposition 7.2. Since the !H (h) sum a(0) + a(1) = h=1 αη (h) is constant, Proposition 7.3 shows in particular that an asset price decreases on average with an increasing market share of the chartists a(1) . From Jury’s test (see Elaydi 1996, p. 181) it is known that a necessary condition for the zeros of χk to lie inside the unit circle is χk (1) > 0 (1)

and

(−1)L+1 χk (−1) > 0.

(L)

If δk = . . . = δk are equal to some δk , then a routine calculation shows that this is the case precisely when a(1) (Lδk a(0)

− R) > R − 1.

Both of the above cases describe scenarios with a stable price process along which mediator 0 has rational expectations in the above sense, whereas the beliefs of mediator 1, in general, will be non-rational and, in fact, could be arbitrarily wrong. As soon as the coeﬃcient matrix A has eigenvalues outside the unit disk, the system (7.41) is unstable and bubble paths will emerge. It follows immediately from B¨ ohm & Chiarella (2005, Thm. 3.2) that such an unstable situation occurs for R > 1 and a(1) suﬃciently close to 0. The somewhat puzzling conjecture for the general nonlinear case is that the presence of suﬃciently many non-rational mediators such as chartists may have a stabilizing eﬀect on the price process. Numerical simulations of the model with varying group sizes of households seem to indicate that for a robust set of parameters, the chartists have non-zero market shares, cf. B¨ ohm & Wenzelburger (2005) and Horst & Wenzelburger (2005).

7.7 Adaptive Learning With Heterogeneous Beliefs

161

7.7 Adaptive Learning With Heterogeneous Beliefs The discussion in Section 7.5 revealed that in order to apply perfect forecasting rules, mediator 0 has to know the excess demand function of all market participants. In the general case, this function will be nonlinear and its estimation requires nonparametric methods provided in Chapter 5. However, in the simple case in which the non-rational mediators i > 0 use simple linear forecasting rules, the estimation problem reduces to a linear one, provided that the switching behavior of households is correctly anticipated. We will ﬁrst outline the general idea and then return to the particularly simple example with two mediators. 7.7.1 The General Idea In this section, we introduce a general principle of estimating an unbiased forecasting rule from historical data. Consider to this end the aggregate (percapita) excess demand function (7.46) of all mediators except mediator 0. Using the notation of Section 7.1, let I

(i) (i) (i) (i) (i) (i) Φ(i) (ηt , qt , Vt , p) + ξt − x ¯ Φ ξt , (ηt , qt , Vt )Ii=1 , p :=

(7.46)

i=1

denote the aggregate excess demand of all mediators except mediator 0. Assume that each mediator i > 0 uses forecasting rules ψ (i) and ϕ(i) of the functional form (i)

qt = ψ (i) (qt−1 , . . . , qt−L )

and

(i)

Vt

= ϕ(i) (qt−1 , . . . , qt−L ).

Inserting these forecasting rules into (7.46) and suppressing the respective functional expressions for notational convenience, the excess demand (7.46) takes the form

(i) Φ ξt , (ηt )Ii=1 , qt−1 , . . . , qt−L , p . (7.47) (0)

Let xt denote the portfolio held by mediator 0 after trading in period t. Then the market-clearing condition (7.4) implies that (0)

−xt

(i) for each t ∈ N. = Φ ξt , (ηt )Ii=1 , qt−1 , . . . , qt−L , pt (0)

(7.48)

Observe that the prices and the portfolios xt appearing in (7.48) are ob(i) servable quantities, whereas the market shares (ηt )Ii=1 are not observable for mediator 0. Moreover, the concrete functional form (7.47) including the involved forecasting rules are unknown to mediator 0. Assume for a moment that a good approximation for the discrete choice model (7.27) with unknown intensity of choice parameters has been developed along with a model for the exogenous noise process {ξt }t∈N . Suppose that

162

7 Heterogeneous Beliefs in a Financial Market (i)

(i)

these models yield suﬃciently good estimates ηˆt for ηt , i = 1, . . . , I and ξˆt for ξt , respectively. Then the following adaptive estimation procedure may be applied. At an arbitrary date t, ﬁrst estimate the functional relationship (0) (i) (7.48) from past observed or estimated data {xs , ξs , (ˆ ηs )Ii=1 , ps , qs }t−1 s=0 . This . yields an approximation Φt of (7.47) such that

(i) .t ξˆt , (ˆ ηt )Ii=1 , qt−1 , . . . , qt−L , p (7.49) Φ describes the approximate excess demand function for assets in period t. Sec(0) (0) ond, let (qt , Vt ) denote the beliefs of mediator 0 as before. Replacing Φ in (7.48) with (7.49) and solving for the ex-dividend price, we would get an approximate temporary equilibrium map

(i) (0) (0) . t ξˆt , (ˆ ηt )Ii=1 , qt−1 , . . . , qt−L , qt , Vt , (7.50) pˆt = G provided that a solution exists. Third, in view of (7.8), taking expectations (0) the condition for qt−1 being unbiased is

! (0) (i) (0) (0) .t ξˆt , (ˆ + dt = qt−1 . ηt )Ii=1 , qt−1 , . . . , qt−L , qt , Vt (7.51) Et−1 G (0)

Hence any solution qt to (7.51) deﬁnes a approximation of the unbiased forecasting rule (7.9) in period t. Up to the two invertibility conditions in (7.50) and (7.51) which may be violated during the estimation process, the forecasting scheme described in this section can, in principle, be carried out at each point in time using the estimation techniques of Chapter 5. The next section demonstrates the success of such an approach in the case with linear mean-variance preferences and linear forecasting rules. 7.7.2 The Case With Two Mediators Consider now the case of two mediators treated in Section 7.6 in which households switch between mediators according to (7.27). The proposed learning scheme now takes a particularly simple form. Assume to this end that the estimates of the market shares are suﬃciently good such that we can abstract from prediction errors of the distribution of households. This assumption seems not (h0) (h1) + ηt = η (h) is constant over time to be too strong because the sum ηt (h0) is known to mediator 0. On the other hand, for all 1 ≤ h ≤ H and each ηt for the unknown choice intensities in (7.27) there are maximum-likelihood estimators available which are consistent under standard regularity assumptions. In the present case of our discrete choice model of the Logit type, these estimators are found in Judge, Griﬃths, Hill, L¨ utkepohl & Lee (1985, Chap. 18, p. 765). The estimators for the choice intensities yield estimates (h1) (h1) ηˆt , h = 1, . . . , H for the fractions of households ηt , h = 1, . . . , H employing mediator 1. The informational constraints faced by mediator 0 are now summarized as follows.

7.7 Adaptive Learning With Heterogeneous Beliefs

163

Assumption 7.2. The information of mediator 0 encompasses the following: (1)

(i) The realized returns Rt of mediator 1 as given by (7.25) are observable quantities; a discrete choice model for the switching behavior of households has been estimated, consistent estimates of the fraction of households are known, such that, for simplicity of exposition, prediction errors are ne(h1) (h1) = ηt , h = 1, . . . , H. glected and ηˆt (ii) The total amount of retradeable assets per capita x¯ is known; (iii) The correct functional form of mediator 1’s forecasting rules is known; these consist of a simple technical trading rule of the form (7.40) and a constant subjective covariance matrix. To specify the noise traders, we impose the following so-called identiﬁability assumption (see Chapter 3, Assumption 3.1) which governs the transactions of the noise traders. Assumption 7.3. The noise-trader behavior is driven by an MA(N )-process {ξt }t∈N , given by N C (n) t−n , ξt = t + n=1

where {t }t∈N is an unobservable iid process on the probability space (Ω, F, P) with zero means E[t ] = 0 which satisﬁes the following: (1) The process {t }t∈N is bounded and, in particular, supt≥1 E [ t γ ] < ∞ for some γ > 2. (2) C (1) , . . . , C (N ) , N ≥ 0 are K × K matrices and the matrix polynomial Γ (z) = IK + C (1) z + · · · + C (N ) z N , z ∈ C is strictly positive real, that is, (i) det Γ (z) = 0 for all z ∈ C with |z| ≤ 1 and (ii) for each z ∈ C with |z| ≤ 1, the matrix Γ (z) + Γ (¯ z ), with z¯ ∈ C denoting the complex conjugate of z, is strictly positive deﬁnite such that

z ) ζ > 0 for all 0 = ζ ∈ RK . ζ Γ (z) + Γ (¯ Adopting Assumption 7.3, we have Et−1 [ξt ] =

N

C (n) t−n

and Vt−1 [ξt ] = V[t ].

n=1

Notice, moreover, that Assumption 7.3 includes the case in which {ξt }t∈N is iid (Set N = 0). By Assumption 7.2, mediator 0 is left to estimate the excess demand (7.49) of mediator 1. A major prerequisite for the success of the estimation is Assumption 7.3 which presumes that the noise trader behavior is not too arbitrary. Clearly, there is no economic justiﬁcation why noise trader should behave according to this presumption other than without it, our learning scheme

164

7 Heterogeneous Beliefs in a Financial Market

may fail to converge.7 By Assumption 7.2, mediator 0 has correct knowledge about the functional form of the forecasting rule (7.40) used by mediator 1 (0) but not its concrete parameterization. Set xt = x ¯ − xt to adjust mediator 0’s portfolio for the total amount of available assets. Then the market-clearing condition (7.48) for period t reads xt =

H

(h1)

ηt α(h)

V (1)−1

L

D(l) qt−l − Rpt + t +

C (n) t−n .

(7.52)

n=1

l=1

h=1

N

Setting θ = (A(11) , . . . , A(HL) , B (1) , . . . , B (H) , C (1) , . . . , C (N ) ) with A(hl) =

1 V (1)−1 D(l) , α(h)

B (h) =

−R V (1)−1 α(h)

(7.53)

(7.54)

for all h = 1, . . . , H, l = 1, . . . , L and

(11) (H1) (11) (H1) yt = ηt qt−1 , , . . . , ηt qt−L , ηt p pt , t , . . . , ηt t−1 , . . . , t−N (7.55) the market-clearing condition (7.52) may be rewritten as xt = θ yt + t .

(7.56)

We use the representation (7.56) to estimate the unknown coeﬃcient matrix θ by applying the approximate-maximum-likelihood (AML) algorithm introduced in Chapter 3, Section 3.3. This recursive scheme generates the successive estimates (11) (HL) ˆ (1) ˆ (H) , Cˆ (1) , . . . , Cˆ (N ) ), , Bt , . . . , B θˆt = (Aˆt , . . . , Aˆt t t t

(7.57)

for θ based upon information available at date t. Since the noise process {t }t∈N is assumed to be unobservable, for each t ∈ N set ˆt := xt − θˆt yˆt ,

(11) (H1) (11) (H1) yˆt := ηt qt−1 , . . . , ηt qt−L , ηt p pt , ˆt−1 , . . . , ˆ , t , . . . , ηt t−N where the ˆt are also called a posteriori prediction errors. The AML algorithm is recursively deﬁned by

8

7

As already pointed out in Chapter 3, identiﬁability assumptions as stated in Assumption 7.3 are fundamental in the systems and control literature. Ljung, S¨ oderstr¨ om & Gustavsson (1975) show that without such identiﬁability assumptions, identiﬁcation methods such as ordinary least squares may fail. 8 We adjusted the dating according to our needs and made use of the matrix inversion lemma (e.g., see Caines 1988).

7.7 Adaptive Learning With Heterogeneous Beliefs

ˆ θt+1 = θˆt + (xt − θˆt yˆt )ˆ yt Pt , = Pt−1 − Pt−1 (ˆ yt yˆt )Pt−1 (1 + yˆt Pt−1 yˆt )−1 ,

Pt

165

(7.58)

for arbitrary initial conditions P−1 and θˆ0 . The estimated coeﬃcients (7.57) obtained from applying (7.58) can now be used to construct an approximation of the perfect forecasting rules deﬁned in (7.32) and (7.37) based on information available at date t. An approximation of (7.32) is deﬁned by (0)

qt

(0)

= ψ.t

(0)

(h1) H )h=1 , θˆt (0)−1 (0) at Vt x ˆt , (0)

(qt−1 − dt ), Vt ; (ηt

(0)

:= R(qt−1 − dt ) −

(7.59)

where xˆt =

L H

(h1)

ηt

(hl) Aˆt qt−l +

h=1 l=1

H

(h1)

ηt

ˆt(h) (q (0) − dt ) + B t−1

N

(n) ¯ Cˆt ˆt−n − x

n=1

h=1

denotes the estimated excess demand of shares in period t corresponding to !N ˆ (n) (7.49). Observe that ξˆt := ˆt−n is an approximation for the exn=1 Ct pected transactions of the noise traders Et−1 [ξt ]. In order to obtain an approximation of the perfect forecasting rule for second moments (7.37), assume that Vt−1 [ξt ] ≡ σξ IK with known σξ .9 It !H (h1) ˆ (h) (1) is an approximation for at V (1)−1 . follows from (7.54) that R1 h=1 ηt B t Setting 3 −1 H (0) (0) (0)−1 (h1) ˆ (h) ηt Bt , Vt := Rat σξ Vt−1 − h=1

an approximation of (7.37) is then given by (0)

Vt

(0) (0) (h1) ˆ = ϕ .t Vt−1 ; (ηt )H h=1 , θt (0)−1 (0) Vt if Vt is positive deﬁnite, := (0) otherwise. Vt−1

(7.60)

The series of forecasts (7.59) and (7.60) generated by (7.58) deﬁnes a AMLbased adaptive learning scheme for mediator 0. Together with the chartist’s forecasts (7.40), the temporary equilibrium map (7.23), and the logit function (7.27), then prices and forecasts under this learning scheme evolve time according to the following set of equations:

9

This assumption is made for simplicity. Since noise traders’ transactions are observable, estimations of Vt−1 [ξt ] could, in principle, be obtained by standard methods.

166

7 Heterogeneous Beliefs in a Financial Market

⎧ ⎪ qt ⎪ ⎪ ⎪ ⎪ ⎪ (1) ⎪ Vt ⎪ ⎪ ⎪ ⎪ ⎪ (1) ⎪ ⎪ qt ⎪ ⎪ ⎨ (0) Vt ⎪ ⎪ ⎪ ⎪ (0) ⎪ qt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θˆt+1 ⎪ ⎪ ⎪ ⎩ Pt

(i)

(i)

(i)

= G(ξt , (at , qt , Vt )1i=0 ) + dt , (1)

= V0 , !L = l=1 D(l) qt−l , (h1) (0) (0) ˆ =ϕ .t Vt−1 ; (ηt )H h=1 , θt , (0) (0) (0) (h1) ˆ = ψ.t (qt−1 − dt ), Vt ; (ηt )H h=1 , θt ,

(7.61)

= θˆt + (xt − θˆt yˆt )ˆ yt Pt , = Pt−1 − Pt−1 (ˆ yt yˆt )Pt−1 (1 + yˆt Pt−1 yˆt )−1 .

The proof that the learning scheme converges to rational expectations for mediator 0 in the sense that ﬁrst and second moments are correct along each orbit of the system depends on whether or not the AML algorithm (7.58) generates strongly consistent estimates, that is, estimates θˆt which converge Pa.s. to θ. Observe that the requirement Vt−1 [ξt ] ≡ σξ IK is needed only for the approximation of the forecasting rule for second moments (7.37). Moreover, the adaptive learning scheme (7.58) is meaningful only if the dynamics under rational expectations for mediator 0 as well as under the scheme itself is stable. Before we state conditions under which strongly consistent estimates obtain, recall that λmin (B) and λmax (B) denote the minimal and maximal eigenvalue of a symmetric matrix B, respectively. Theorem 7.2. Assume that the following hypotheses are satisﬁed. (i) The noise process {ξt }t∈N satisﬁes Assumption 7.3 with known Vt−1 [ξt ] = σξ IK and the dividend process {dt }t∈N satisﬁes Assumption 7.1 (ii). (ii) Mediator 1 uses a linear rule (7.40) and constant covariance matrices as forecasting rules. (iii) The informational constraints of mediator 0 satisfy Assumption 7.2. (iv) The price process (7.23) is stable under rational expectations for mediator 0 as given in (7.38) and under the application of the AML-based learning scheme (7.58) for mediator 0 as given by (7.61), in the sense that for both cases the respective processes of of prices and forecast remain bounded, i.e., (0) (1)

(qt , qt , qt ) < M P − a.s. for all t ∈ N for some constant M . If the stochastic process {yt }t∈N deﬁned in (7.55) satisﬁes the weak excitation condition $! %% $ t log e + λmax s=1 ys ys $! % → 0 P − a.s. as t → ∞, (7.62) t λmin s=1 ys ys

7.7 Adaptive Learning With Heterogeneous Beliefs

167

then the sequence of estimates {θˆt }t∈N generated by the AML algorithm (7.58) is strongly consistent, that is θˆt → θ

P − a.s. as t → ∞.

The proof follows directly from Chapter 3, Proposition 3.3. As in Chapter 3, the result is independent of initial conditions as long as the system is stable under the learning scheme. Combining the results of Chapter 3 with Proposition 3.4 in Horst & Wenzelburger (2005), it is intuitively clear how the forecasts could be ‘censored’ in such a way that the system remains stable under learning. The general idea of such a censoring was discussed in Chapter 5. The major obstacle for convergence is the weak excitation condition (7.62). In feedback systems such as (7.38), it is well-known that a violation of condition (7.62) may result in a failure of consistent estimates and thus in a failure of the learning scheme. However, as discussed in Chapter 3, this problem can be resolved by occasionally perturbing the forecasts such that (7.62) is satisﬁed. Moreover, the success of the learning scheme depends essentially on the success of estimating the excess demand function. This estimation by means of the AML algorithm can only be successful if, apart from the switching behavior of households, the excess demand function is linear. Non-linear excess demand functions as, for example, in Brock & Hommes (1997b) will generally require non-linear estimation techniques. For the present model, this problem arises naturally as soon as chartists use empirical covariance matrices. However, the concept of estimating the excess demand readily carries over to the non-linear case. Then non-parametric techniques as presented in Chapter 5 can be applied to estimate the excess demand function.

Concluding Remarks The analysis of adaptive learning in a ﬁnancial market with heterogeneous traders showed that good forecasts hinge essentially on good estimates of the demand behavior of market participants. It was demonstrated that expectations should be based on estimates of the excess demand function of traders. The proposed learning scheme made use of the simple observation that the excess demand function is linear when households are characterized by linear mean-variance preferences and mediators by a simple linear forecasting technology. The remaining nonlinearity rests on the switching behavior of households for which consistent maximum likelihood estimators exist. In a special case with only one chartist, it was shown that this methodology leads to a learning scheme which converges under econometric assumptions that are standard in the literature on estimation and optimal control. While situations in which an agent successfully learns to have rational expectations are likely to remain special cases, one should proceed with caution when assessing the plausibility of a learning scheme. As shown by Jungeilges

168

7 Heterogeneous Beliefs in a Financial Market

(2003), the underlying linearity hypothesis for applying the ordinary-leastsquares learning scheme in nonlinear models may be rejected within ﬁnite time by means of techniques from linear statistics. Apart from the convergence issue, the main conclusion of this chapter is that a plausible learning scheme should take into account the correct trading structure of the market. The potential of this line of research may be outlined as follows. First, the learning scheme should be extended to an algorithm which uses on-line estimations of the distributions of households instead of their true values. Second, non-parametric estimation techniques should be applied to identify forecasting rules for models with nonlinear excess demand functions. However, the biggest challenge is to abandon the assumption of price-taking behavior. The structure of an unbiased forecasting rule points at a possibility to exploit the knowledge of excess demand functions strategically. This will be carried out in Wenzelburger (2005).

7.8 Mathematical Appendix Proof of Proposition 7.1. We preface the proof of the proposition by a technical lemma. Lemma 7.3. Let B, C be arbitrary symmetric positive deﬁnite matrices. Then √ 3 √ √ √ ( B C B) D := B −1 B −1 is a symmetric positive deﬁnite matrix with C = DBD.

√ Proof. √ Since √ B is symmetric and positive deﬁnite, its square root B, where B = √B B, is√again symmetric and positive deﬁnite and hence invertible with ( B)−1 = B −1 . It is straightforward √ to verify that D as deﬁned above √ satisﬁes C = DBD. Since C, B, and B −1 are all symmetric and positive deﬁnite, it is veriﬁed directly that D is symmetric and positive deﬁnite.

By Lemma 7.3, Πt as given by (7.36) is a solution to the equation Πt−1 Λt−1 Πt−1 = R2 Vt−1 . (0)

!I (i) (i)−1 If Πt − i=1 at Vt is positive deﬁnite, then (7.37) is well deﬁned and the assertion follows from (7.35).

Proof of Lemma 7.1. The ﬁrst statement follows from H η (h) (0)−1 −1 Vt At = α(h) h=1

(0)

and Lemma 7.3 applied to (7.35). The second one is a special case with Vt (0) Vt−1 for all times t.

≡

7.8 Mathematical Appendix

169

(0)

Proof of Lemma 7.2. We seek a symmetric matrix Vt which satisﬁes (7.35) and which stays positive deﬁnite for all times t. Under the given hypotheses, (0) this is equivalent to the existence of a symmetric and positive matrix Vt with 3 I σξ (0) (0)−1 (i) (i)−1 (0)−1 Vt−1 − at Vt = at Vt . (7.63) R i=1 (0)

Choose V0 as in (iii). Then (7.63), the deﬁnition of the square root, and induction imply that (0)−1

Vt where (0k)

λt

=

= O diag(λt

(01)

3 I (i) at (ik) (0k) λ , λ − t−1 (0) (0) Rat i=1 at σξ

(0K)

, . . . , λt

)O,

k = 1, . . . , K.

(7.64)

Equation (7.64) is a list of K strictly monotonically increasing and strictly 4 concave scalar maps of the form λt = b1 λt−1 − b2 with random coeﬃcients (0) b1 and b2 . These determine the eigenvalues of the covariance matrix Vt . (i) Condition (ii) now guarantees that for ﬁxed at , i = 0, . . . , I, each of the maps (7.64) has two positive deterministic ﬁxed points, the larger one of which is asymptotically stable. Hence, there exists a compact set K ⊂ RK ++ which is (01) (0K) forward invariant under the maps (7.64) such that (λt , . . . , λt ) ∈ K for all times t ≥ 0.

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Vol. 523: W.-B. Zhang, A Theory of Interregional Dynamics. XI, 231 pages. 2003. Vol. 524: M. Frölich, Programme Evaluation and Treatment Choise. VIII, 191 pages. 2003. Vol. 525: S. Spinler, Capacity Reservation for Capital- Intensive Technologies. XVI, 139 pages. 2003. Vol. 526: C. F. Daganzo, A Theory of Supply Chains. VIII, 123 pages. 2003. Vol. 527: C. E. Metz, Information Dissemination in Currency Crises. XI, 231 pages. 2003.

Vol. 507: M. Koksalan, S. Zionts, Multiple Criteria Decision Making in the New Millenium. XII, 481 pages. 2001.

Vol. 528: R. Stolletz, Performance Analysis and Optimization of Inbound Call Centers. X, 219 pages. 2003.

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Vol. 529: W. Krabs, S. W. Pickl, Analysis, Controllability and Optimization of Time-Discrete Systems and Dynamical Games. XII, 187 pages. 2003.

Vol. 509: D. Hornung, Investment, R&D, and Long-Run Growth. XVI, 194 pages. 2002.

Vol. 530: R. Wapler, Unemployment, Market Structure and Growth. XXVII, 207 pages. 2003.

Vol. 531: M. Gallegati, A. Kirman, M. Marsili (Eds.), The Complex Dynamics of Economic Interaction. XV, 402 pages, 2004. Vol. 532: K. Marti, Y. Ermoliev, G. Pﬂug (Eds.), Dynamic Stochastic Optimization. VIII, 336 pages. 2004. Vol. 533: G. Dudek, Collaborative Planning in Supply Chains. X, 234 pages. 2004. Vol. 534: M. Runkel, Environmental and Resource Policy for Consumer Durables. X, 197 pages. 2004. Vol. 535: X. Gandibleux, M. Sevaux, K. Sörensen, V. Tkindt (Eds.), Metaheuristics for Multiobjective Optimisation. IX, 249 pages. 2004.

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Vol. 536: R. Brüggemann, Model Reduction Methods for Vector Autoregressive Processes. X, 218 pages. 2004.

Vol. 563: A. Seeger (Ed.), Recent Advances in Optimization. XI, 455 pages. 2006.

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Vol. 564: P. Mathieu, B. Beauﬁls, O. Brandouy (Eds.), Artiﬁcial Economics. XIII, 237 pages. 2005.

Vol. 538: S. Kokot, The Econometrics of Sequential Trade Models. XI, 193 pages. 2004.

Vol. 565: W. Lemke, Term Structure Modeling and Estimation in a State Space Framework. IX, 224 pages. 2006.

Vol. 539: N. Hautsch, Modelling Irregularly Spaced Financial Data. XII, 291 pages. 2004.

Vol. 566: M. Genser, A Structural Framework for the Pricing of Corporate Securities. XIX, 176 pages. 2006.

Vol. 540: H. Kraft, Optimal Portfolios with Stochastic Interest Rates and Defaultable Assets. X, 173 pages. 2004.

Vol. 567: A. Namatame, T. Kaizouji, Y. Aruga (Eds.), The Complex Networks of Economic Interactions. XI, 343 pages. 2006.

Vol. 541: G.-y. Chen, X. Huang, X. Yang, Vector Optimization. X, 306 pages. 2005. Vol. 542: J. Lingens, Union Wage Bargaining and Economic Growth. XIII, 199 pages. 2004. Vol. 543: C. Benkert, Default Risk in Bond and Credit Derivatives Markets. IX, 135 pages. 2004. Vol. 544: B. Fleischmann, A. Klose, Distribution Logistics. X, 284 pages. 2004. Vol. 545: R. Hafner, Stochastic Implied Volatility. XI, 229 pages. 2004. Vol. 546: D. Quadt, Lot-Sizing and Scheduling for Flexible Flow Lines. XVIII, 227 pages. 2004. Vol. 547: M. Wildi, Signal Extraction. XI, 279 pages. 2005. Vol. 548: D. Kuhn, Generalized Bounds for Convex Multistage Stochastic Programs. XI, 190 pages. 2005. Vol. 549: G. N. Krieg, Kanban-Controlled Manufacturing Systems. IX, 236 pages. 2005. Vol. 550: T. Lux, S. Reitz, E. Samanidou, Nonlinear Dynamics and Heterogeneous Interacting Agents. XIII, 327 pages. 2005. Vol. 551: J. Leskow, M. Puchet Anyul, L. F. Punzo, New Tools of Economic Dynamics. XIX, 392 pages. 2005. Vol. 552: C. Suerie, Time Continuity in Discrete Time Models. XVIII, 229 pages. 2005. Vol. 553: B. Mönch, Strategic Trading in Illiquid Markets. XIII, 116 pages. 2005. Vol. 554: R. Foellmi, Consumption Structure and Macroeconomics. IX, 152 pages. 2005. Vol. 555: J. Wenzelburger, Learning in Economic Systems with Expectations Feedback (planned) 2005. Vol. 556: R. Branzei, D. Dimitrov, S. Tijs, Models in Cooperative Game Theory. VIII, 135 pages. 2005.

Vol. 568: M. Caliendo, Microeconometric Evaluation of Labour Market Policies. XVII, 258 pages. 2006. Vol. 569: L. Neubecker, Strategic Competition in Oligopolies with Fluctuating Demand. IX, 233 pages. 2006. Vol. 570: J. Woo, The Political Economy of Fiscal Policy. X, 169 pages. 2006. Vol. 571: T. Herwig, Market-Conform Valuation of Options. VIII, 104 pages. 2006. Vol. 572: M. F. Jäkel, Pensionomics. XII, 316 pages. 2006 Vol. 573: J. Emami Namini, International Trade and Multinational Activity, X, 159 pages, 2006. Vol. 574: R. Kleber, Dynamic Inventory Management in Reverse Logisnes, XII, 181 pages, 2006. Vol. 575: R. Hellermann, Capacity Options for Revenue Management, XV, 199 pages, 2006. Vol. 576: J. Zajac, Economics Dynamics, Information and Equilibnum, X, 284 pages, 2006. Vol. 577: K. Rudolph, Bargaining Power Effects in Financial Contracting, XVIII, 330 pages, 2006. Vol. 578: J. Kühn, Optimal Risk-Return Trade-Offs of Commercial Banks, IX, 149 pages, 2006. Vol. 579: D. Sondermann, Introduction to Stochastic Calculus for Finance, X, 136 pages, 2006. Vol. 580: S. Seifert, Posted Price Offers in Internet Auction Markets, IX, 186 pages, 2006. Vol. 581: K. Marti; Y. Ermoliev; M. Makowsk; G. Pﬂug (Eds.), Coping with Uncertainty, XIII, 330 pages, 2006 (planned). Vol.583: Konnov, Igor V.; Luc, Dinh The; Rubinov, Alexander M. (Eds.), Generalized Convexity and Related Topics, IX, 469 pages, 2006 (planned).